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LORD  KELVIN 


Elements   of 
Natural    Philosophy 


BY 


LORD   KELVIN,  LL.D.,  D.C.L.,  F.R.S. 


AND 


PETER  GUTHRIE  TAIT,  M.A. 


NEW  YORK 
P.    F.    COLLIER    &    SON 

M  C  M  I  I 
23 


Qa\  university  of  CALIFOnNiA 

.       '  .  SANTA  BAliliAKA 


U-' '-- 


k  + 


SCIENCE 


VI  PREFACE 

of  notes  of  a  part  of  the  Glasgow  course,  drawn  up  for 
Sir  W.  Thomson  by  John  Perguson,  Esq.,  and  printed  for 
the  use  of  his  students, 

"We  have  had  considerable  difficulty  in  compiling  this 
treatise  from  the  larger  work — arising  from  the  necessity  for 
condensation  to  a  degree  almost  incompatible  with  the  design 
to  omit  nothing  of  importance:  and  we  feel  that  it  would 
have  given  us  much  less  trouble  and  anxiety,  and  would 
probably  have  ensured  a  better  result^  had  we  written  the 
volume  anew  without  keeping  the  larger  book  constantly 
before  us.  The  sole  justification  of  the  course  we  have  pur- 
sued is  that  wherever,  in  the  .present  volume,  the  student 
may  feel  further  information  to  be  desirable,  he  will  have 
no  difficulty  in  finding  it  in  the  corresponding  pages  of  the 
larger  work. 

"  A  great  portion  of  the  present  volume  has  been  in  type 
since  the  autumn  of  1863,  and  has  been  printed  for  the  use 
of  our  classes  each  autumn  since  that  date." 

To  this  we  would  now  only  add  that  the  whole  has  been 
revised,  and  that  we  have  endeavoured  tc  simplify  those 
portions  which  we  have  found  by  experience  to  present 
difficulties  .to  our  students. 

The  present  edition  has  been  carefully  revised  by^Mr  W. 
BURNSIDE,  of  Pembroke  College  :  and  an  Index,  of  which  we 
have  recognized  the  necessity,  has  been  drawn  up  for  us  by 
Mr  Scott  Lang. 

W.  THOMSON. 
P.  G.  TAIT. 

January,  1879. 


CONTENTS. 

DIVISION  I.     Preliminary. 

PAGE 

Chap.  I.       Kinematics. i 

„  11.  Dynamical  Laws  and  Jc'rinciples       ,        .        56 

„  III,     Experience iii 

„  IV.  Measures  and  Instruments        .        .        .122 

DIVISION  It.     Abstract  Dynamics. 

„        V.      Introductory c  136 

„        VI.     Statics  of  a  Particle.     Attraction  .        .  140 

,,      VII.    Statics  of  Solids  and  Fluids    .        .        .  199 

Appendix 282 


DIVISION  I. 

PRELIMINARY. 

CHAPTER    I.— KINEMATICS. 

1.  The  science  which  investigates  the  action  of  Force  is  called,  by 
the  most  logical  writers,  Dynamics.  It  is  commonly,  but  erroneously, 
called  Mechanics  ;  a  term  employed  by  Newton  in  its  true  sense, 
the  Science  of  Machines,  and  the  art  of  making  them. 

2.  Force  is  recognized  as  acting  in  two  ways  : 

1°  so  as  to  compel  rest  or  to  prevent  change  of  motion,  and 

2"  so  as  to  produce  or  to  chang6  motion. 
Dynamics,  therefore,  is  divided  into  two  parts,  which  are  conveniently 
called  Statics  and  Kinetics. 

3.  In  Statics  the  action  of  force  in  maintaining  rest,  or  preventing 
change  of  motion,  the  'balancing  of  forces,'  or  Equilibrium,  is 
investigated ;  in  Kinetics,  the  action  of  force  in  producing  or  ia 
changing  motion. 

4.  In  Kinetics  it  is  not  mere  motion  which  is  investigated,  but  the 
•relation  of  forces  to  motion.  The  circumstances  of  mere  motion, 
considered  without  reference  to  the  bodies,  moved,  or  to  the  forces 
producing  the  motion,  or  to  the  forces  called  into  action  by  the 
motion,  constitute  the  subject  of  a  branch  of  Pure  Mathematics, 
which  is  called  Kinem.\tics,  or,  in  its  more  practical  branches, 
Mechanism. 

5.  Observation  and  experiment  have  afforded  us  the  means  of 
translating,  as  it  were,  from  Kinematics  into  Dynamics,  and  vice  versA. 
This  is  merely  mentioned  now  in  order  to  show  the  necessity  for, 
and  the  value  of,  the  preliminary  matter  we  are  about  to  introduce. 

6.  Thus  it  appears  that  there  are  many  properties  of  motion, 
displacement,  and  deformation,  which  may  be  considered  altogether 
independently  of  force,  mass,  chemical  constitution^  elasticity,  tempe- 
rature, magnetism,  electricity  ;  and  that  the  preliminary  consideration 
of  such  properties  in  the  abstract  is  of  very  great  use  for  Natural 


2  PRELIMINARY. 

Philosophy.  We  devote  to  it,  accordingly,  the  whole  of  this  chapter  j 
which  will  form,  as  it  were,  the  Geometry  of  the  subject,  embracing 
what  can  be  observed  or  concluded  with  regard  to  actual  motions, 
as  long  as  the  cause  is  not  sought.  In  this  category  vve  shall  first  take 
up  the  free  motion  of  a  point,  then  the  motion  of  a  point  attached 
to  an  inextensible  cord,  then  the  motions  and  displacements  of  rigid 
systems — and  finally,  the  deformations  of  solid  and  fluid  masses. 

7.  WTien  a  point  moves  from  one  position  to  another  it  must 
evidently  describe  a  continuous  line,  which  may  be  curved  or  straight, 
or  even  made  up  of  portions  of  curved  and  straight  lines  meeting 
each  other  at  any  angles.  If  the  motion  be  that  of  a  material  particle, 
however,  there  can  be  no  abrupt  change  of  velocity,  nor  of  direction 
unless  where  the  velocity  is  zero,  since  (as  we  shall  afterwards  see) 
such  would  imply  the  action  of  an  infinite  force.  It  is  useful  to  con- 
sider at  the  outset  various  theorems  connected  with  the  geometrical 
notion  of  the  path  described  by  a  moving  point ;  and  these  we  shall 
now  take  up,  deferring  the  consideration  of  Velocity  to  a  future 
section,  as  being  more  closely  connected  with  physical  ideas. 

8.  The  direction  of  motion  of  a  moving  point  is  at  each  instant 
the  tangent  drawn  to  its  path,  if  the  path  be  a  curve ;  or  the  path 
itself  if  a  straight  line.  This  is  evident  from  the  definition  of  the 
tangent  to  a  curve. 

9.  If  the  path  be  not  straight  the  direction  of  motion  changes 
from  point  to  point,  and  the  rate  of  this  change,  per  unit  of  length 
of  the  curve,  is  called  the  Cumature.     To  exemplify  this,  suppose 

^  two  tangents,  PT,   QU,  drawn  to  a  circle, 

and  radii  OP^  OQ,  to  the  points  of  contact. 

The    angle    between    the   tangents    is   the 

ijy  change    of   direction    between    P  and    Q, 

and  the  rate  of  change  is  to  be  measured 

by  the  relation  between  this  angle  and  the 

_  length  of  the  circular  arc  PQ.     Now,  if  6 

be  the  angle,  o  the  arc,  and  r  the  radius,  we 

see  at  once  that  (as  the  angle  between  the  radii  is  equal  to  the 

angle  between  the  tangents,  and  as  the  measure  of  an  angle  is  the 

ratio  of  the  ai'c  to  the  radius,  §  54)  rO-s,  and  therefore  -=-  is  the 

measure  of  the  curvature.  Hence  the  curvature  of  a  circle  is  in- 
versely as  its  radius,  and  is  measured,  in  terms  of  the  proper  unit  of 
curvature,  simply  by  the  reciprocal  of  the  radius. 

10.  Any  small  portion  of  a  curve  may  be  approximately  taken 
as  a  circular  arc,  the  approximation  being  closer  and  closer  to  the 
truth,  as  the  assumed  arc  is  smaller.  The  curvature  at  any  point 
is  the  reciprocal  of  the  radius  of  this  circle  for  a  small  arc  on  each 
side  of  the  point. 

11.  If  all  the  points  of  the  curve  lie  in  one  plane,  it  is  called  z plane 
curve,  and  if  it  be  made  up  of  portions  of  straight  or  curved  lines  it 


KINEMATICS.  3 

is  called  '3l  plane  polygon.  If  the  line  do  not  lie  in  one  plane,  we  have 
in  one  case  what  is  called  a  cut-ve  of  double  cun'aiure,  in  the  other 
a  gauche  polygon.  The  term  '  curve  of  double  curvature '  is  a.  very 
bad  one,  and,  though  in  very  general  use,  is,  we  hope,  not  inera- 
dicable. The  fact  is,  that  there  are  not  two  curvatures,  but  only 
a  cur\'ature  (as  above  defined)  of  which  the  plane  is  continuously 
changing,  or  twisting,  round  the  tangent  line.  The  course  of  such 
a  curve  is,  in  common  language,  well  called  '  tortuous  ; '  and  the  mea- 
sure of  the  corresponding  property  is  conveniently  called  Tortuosity. 

12.  The  nature  of  this  will  be  best  understood  by  considering  the 
curve  as  a  polygon  whose  sides  are,  indefinitely  small.  Any  two 
consecutive  sides,  of  course,  lie  in  a  plane — and  in  that  plane  the 
curvature  is  measured  as  above ;  but  in  a  curve  which  is  not  plane 
the  third  side  of  the  polygon  will  not  be  in  the  same  plane  with  the 
first  two,  and  therefore  the  new  plane  in  which  the  curvature  is  to 
be  measured  is  different  from  the  old  one.  The  plane  of  the  cur\-a- 
ture  on  each  side  of  any  point  of  a  tortuous  curve  is  sometimes  called 
the  Osculating  Plane  of  the  curve  at  that  point.  As  two  successive 
positions  of  it  contain  the  second  side  of  the  polygon  above  men- 
tioned, it  is  evident  that  the  osculating  plane  passes  from  one  position 
to  the  next  by  revolving  about  the  tangent  to  the  curve. 

13.  Thus,  as  we  proceed  along  such  a  curve,  the  curvature  in 
general  varies ;  and,  at  the  same  time,  the  plane  in  which  the  cur- 
vature lies  is  turning  about  the  tangent  to  the  curve.  The  rate  of 
torsion,  or  the  tortuosity,  is  therefore  to  be  measured  by  the  rate  at 
which  the  osculating  plane  turns  about  the  tangent,  per  unit  length 
of  the  curve.  The  simplest  illustration  of  a  tortuous  curve  is  the 
thread  of  a  screw.     Compare  §  41  {d). 

14.  The  Inttgral  Curvature,  or  whole  change  of  direction,  of  an  arc 
of  a  plane  curve,. is  the  angle  through  which  the  tangent  has  turned 
as  we  pass  from  one  extremity  to  the  other.  The  average  cun:ature 
of  any  portion  is  its  whole  curvature  divided  by  its  length.  Suppose 
a  line,  drawn  through  any  fixed  point,  to  turn  so  as  always  to  be 
parallel  to  the  direction  of  motion  of  a  point  describing  the  curve  : 
the  angle  through  which  this  turns  during  the  motion  of  the  point 
exhibits  what  we  have  defined  as  the  integral  curvature.  In  esti- 
mating this,  we  must  of  course  take  the  enlarged  modern  meaning 
of  an  angle,  including  angles  greater  than  two  right  angles,  and  also 
negative  angles.  Thus  the  integral  curvature  of  any  closed  curve  or 
broken  line,  whether  everywhere  concave  to  the  interior  or  not,  is  four 
right  angles,  provided  it  does  not  cut  itself.     That  of  a  Lemniscate, 

2    is  zero.     That  of  the  Epicyloid  p)   is  eight  right  angles ;  and 

so  on. 

15.  The' definition  in  last  section  may  evidently  be  extended  to 
a  plane  polygon,  and  the  integral  change  of  direction,  or  the  angle 
between  the  first  and  last  sides,  is  then  the  sum  of  its  exterior  angles, 
all  the  sides   being   produced  each  in   the  direction  in  which  the 


PRELIMINARY. 


moving  point  describes  it  while  passing  round  the  figure.  This  is 
true  whether  the  polygon  be  closed  or  not.  If  closed,  then,  as  long 
as  it  is  not  crossed,  this  sum  is  four  right  angles, — an  extension  o< 
the  result  in  Euclid,  where  all  reentrant  polygons  are  excluded.    In 

the  star- shaped  figure  ^^ ,  it  is  ten  right  angles,  wanting  the  sum  of 

the  five  acute  angles  of  the  figure;  i.e.  it  is  eight  right  angles. 

16.  A  chain,  cord,  or  fine  wire,  or  a  fine  fibre,  filament,  or  hair, 
may  suggest,  what  is  not  to  be  found  among  natural  or  artificial  pro- 
ductions, a  perfectly  flexible  and  inextensible  line.  The  elementary 
kinematics  of  this  subject  require  no  investigation.  The  mathematical 
condition  to  be  expressed  in  any  case  of  it  is  simply  that  the  distance 
measured  along  the  line  from  any  one  point  to  any  other,  remains 
constant,  however  the  line  be  bent. 

17.  The  use  of  a  cord  in  mechanism  presents  us  with  many 
practical  applications  of  this  theory,  which  are  in  general  extremely 
simple;  although  curious,  and  not  always  very  easy,  geometrical 
problems  occur  ii^  connexion  with  it.  We  shall  say  nothing  here 
about  such  cases  as  knots,  knitting,  weaving,  etc.,  as  being  exces- 
sively difficult  in  their  general  development,  and  too  simple  in  the 
ordinary  cases  to  require  explanation. 

18.  The  simplest  and  most  useful  applications  are  to  the  Pulley 
and  its  combinations.     In  theory  a  pulley  is  simply  a  smooth  body 

which  changes  the  direction  of  a  flexible  and  inextensible 
cord  stretched  across  part  of  its  surface ;  in  practice  (to 
escape  as  much  as  possible  of  the  inevitable  friction) 
it  is  a  wheel,  on  part  of  whose  circumference  the  cord 
is  WTapped. 

(i)  Suppose  we  have  a  single  pulley  B,  about  which 
the  flexible  and  inextensible  cord  ABP  is  wrapped,  and 
suppose  its  free  portions  to  be  parallel. 
If  {A  being  fixed)  a  point  P  of  the  cord 
be  moved  to  P' ,  it  is  evident  that  each 
of  the  portions  AB  and  PB  will  be  -. 
shortened  by  one-half  of  PP'.  Hence,  1 
when  P  moves  through  any  space  in 
B  the  direction  of  the  cord,  the  pulley  B 
moves  in  the  same  direction,  through 
half  the  space. 
(2)  If  there  be  two  cords  and  two  pulleys,  the 
ends  AA'  being  fixed,  and  the  other  end  of  AB 
befng  attached  to  the  pulley  B' — then,  if  all  free 
parts  of  the  cord  are  parallel,  when  P  is  moved  to 
jP',  B  moves  in  the  same  direction  through  half  the 
space,  and  carries  with  it  one  end  of  the  cord  AB. 
Hence  B  moves  through  half  the  space  B  did,  that 
is,  one  Joint h  of  PP' 


r 


iP 


iP' 


V^ 


KINEMATICS.  5 

(3)  And  so  on  for  any  number  of  pulleys,  if  they  be  arranged 
in  the  above  manner.  Similar  considerations  enable  us  to  deter- 
mine the  relative  motions  of  all  parts  of  other  systems  of  pulleys  and 
cords  as  long  as  all  the  free  parts  of  the  cords  are  parallel 

Of  course,  if  a  pulley  h^  fixed,  the  motion  of  a  point  of  one  end  of 
the  cord  to  or  from  it. involves  an  equal  motion  of  the  other  end 
Jrom  or  to  it. 

If  the  strings  be  not  parallel,  the  relations  of  a  single  pulley  or 
of  a  system  of  pulleys  are  a  litde  complex,  but  present  no  difficulty. 

19.  In  the  mechanical  tracing  of  curves,  a  flexible  and  inextensible 
cord  is  often  supposed.  Thus,  in  drawing  an  ellipse,  the  focal  pro- 
perty of  the  curve  shows  us  that  if  we  fix  the  ends  of  such  a  cord 
to  the  foci  and  keep  it  stretched  by  a  pencil,  the  pencil  will  trace 
the  curve. 

By  a  ruler  moveable  about  one  focus,  and  a  string  attached  to  a 
point  in  the  ruler  and  to  the  other  focus,  and  kept  tight  by  a  pencil 
sliding  along  the  edge  of  the  ruler,  the  hyperbola  may  be  described 
by  the  help  of  its  analogous  focal  property ;  and  so  on. 

20.  But  the  consideration  of  evolutes  is  of  some  importance  in 
Natural  Philosophy,  especially  in  certain  mechanical  and  optical 
questions,  and  we  shall  therefore  devote  a  section  or  two  to  this 
application  of  Kinematics. 

Def.  If  a  flexible  and  inextensible  string  be  fixed  at  one  point 
of  a  plane  curve,  and  stretched  along  the  curve,  and  be  then 
unwound  in  the  plane  of  the  curve,  every  point  of  it  will  describe 
an  Involute  of  the  curve.  The  original  curve  is  called  the  Evolute  of 
any  one  of  the  others. 

21.  It  will  be  observed  that  we  speak  of  an  involute,  and  of  the 
evolute,  of  a  curve.  In  fact,  as  will  be  easily  seen,  a  curve  can  have 
but  one  evolute,  but  it  has  an  infinite  number  of  involutes.  For  all 
that  we  have  to  do  to  vary  an  involute,  is  to  change  the  point  of 
the  curve  from  which  the  tracing-point  starts,  or  consider  the  invo- 
lutes described  by  different  points  of  the  string ;  and  these  will,  in 
general,  be  different  curves.  But  the  following  section  shows  that 
there  is  but  one  evolute. 

22.  Let  AB  be  any  curve,  PQ,  a  portion  of  an  involute,  pF,  qQ 
^positions  of  the  free  part  of  the  string.  It  will  be  seen  at  once 
that  these  must  be  tangents  to  die  arc 
AB  at  p  and  q.  Also  the  string  at 
any  stage,  as  pP,  ultimately  revolves 
about/.  Hence /i' is  «^r;//^/ (or  per- 
pendicular to  the  tangent)  to  the  curve 
PQ.  And  thus  the  evolute  of  PQ  is 
a  definite  curve,  viz.  the  envelop  of  (or 
line  which  is  touched  by)  the  normals  drawn  at  ever)'  point  of  PQ, 
or,  which  is  the  same  thing,  the  locus  of  the  centres  of  the  circles 
which  have  at  each  point  the  same  tangent  and  curvature  as  tha 
curve  PQ.     And  we  may  merely  mention,  as  an  obvious  result  of  thci 


6  PRELIMINARY, 

mode  of  tracing,  that  the  arc  qp  is  equal  to  the  difference  of  ^^and* 
pPy  or  that  the  zxcpA  is  equal  to/i*.     Compare  §  104- 

23.  The  rate  of  motion  of  a  point,  or  its  rate  of  change  of  position, 
is  called  its  Velocity.  It  is  greater  or  less  as  the  space  passed  over 
in  a  given  time  is  greater  or  less :  and  it  may  be  uniform^  i.  e..  the 
same  at  every  instant ;  or  it  may  be  variable. 

Uniform  velocity  is  measured  by  the  space  passed  over  in  unit  of 
time,  and  is,  in  general,  expressed  in  feet  or  in  metres  per  second; 
if  very  great,  as  in  the  case  of  light,  it  may  be  measured  in  miles  per 
second.  It  is  to  be  observed  that  Time  is  here  used  in  the  abstract 
sense  of  a  uniformly-increasing  quantity — what  in  the  differential  cal- 
culus is  called  an  independent  variable.  Its  physical  definition  is 
given  in  the  next  chapter. 

24.  Thus  a  point,  which  moves  uniformly  with  velocity  v,  describes 
a  space  of  v  feet  each  second,  and  therefore  vt  feet  in  /  seconds, 
/  being  any  number  whatever.  Putting  s  for  the  space  described 
in  /  seconds,  we  have  s  -  vt. 

Thus  with  unit  velocity  a  point  describes  unit  of  space  in  unit  of 
time. 

25.  It  is  well  to  observe  here,  that  since,  by  our  formula,  we  haVe 
generally  s 

and  since  nothing  has  been  said  as  to  the  magnitudes  of  s  and  t,  we 
may  take  these  as  small  as  we  choose.  Thus  we  get  'the  same  result 
whether  we  derive  v  from  the  space  described  in  a  million  seconds,  or 
from  that  described  in  a  millionth  of  a  second.  This  idea  is  very  useful, 
as  it  makes  our  resuUs  intelligible  when  a  variable  velocity  has  to  be 
measured,  and  we  find  ourselves  obliged  to  approximate  to  its  value 
(as  in  §  28)  by  considering  the  space  described_in  an  interval  so 
short,  that  during  its  lapse  the  velocity  does  not  sensibly  alter  in  value. 

26.  When  the  point  does  not  move  uniformly,  the  velocity  is 
variable,  or  different  at  different  successive  instants :  but  we  define 
the  average  velocity  during  any  time  as  the  space  described  in  that 
time,  divided  by  the  time ;  and,  the  less  the  interval  is,  the  more 
nearly  does  the  average  velocity  coincide  with  the  actual  velocity  at 
any  instant  of  the  interval.  Or  again,  we  define  the  exact  velocity  at 
any  instant  as  the  space  which  the  point  would  have  described  in  one 
second,  if  for  such  a  period  it  kept  its  velocity  unchanged. 

27.  That  there  is'  at  every  instant  a  definite  velocity  for  any  moving 
point,  is  evident  to  all,  and  is  matter  of  everyday  conversation.  Thus, 
a  railway  train,  after  starting,  gradually  increases  its  speed,  and  every 
one  understands  what  is  meant  by  saying  that  at  a  particular  instant  it 
moves  at  the  rate  of  ten  or  of  fifty  miles  an  hour, — although,  in  the 
course  of  an  hour,  it  may  not  have  moved  a  mile  altogether.  ^  We 
may  suppose  that,  at  any  instant  during  the  motion,  the  steam  is  so 
adjusted  as  to  keep  the  train  running  for  some  time  at  a  uniform 
velocity.     This  is  the  velocity  which  the  train  had  jxX  the  instant  in 


KINEMATICS.  7 

question.  Without  supposing  any  such  definite  adjustment  of  the 
driving-power  to  be  made,  we  can  evidently  obtain  an  approximation 
to  the  velocity  at  a  particular  instant,  by  considering  (§  25)  the 
motion  for  so  short  a  time,  that  during  that  time  the  actual  variation 
of  speed  may  be  small  enough  to  be  neglected. 

i  .  ■,     , 

28.  In  fact,  if  v  be  the  velocity  at  either  begmning  or  end,  or  at 

any  instant,  of  an  interval  /,  and  s  the  space  actually  described  in 

that  interval j  the  equation  v=-- .(which  expresses  the  definition  of 

the  average  velocity,  §*6)  is  more  and  more  nearly  true,  as  the 
velocity  is  more  nearly  uniform  during  the  interval  /;  so  that  if  we 
take  the  interval  small  enough  the  equation  may  be  made  as  nearlv 
exact  as  we  choose.    Thus  the  set  of  values — 

Space  described  in  one  second, 

Ten  times  the  space  described  in  the  first  tenth  of  a  second, 

A  hundred  „  „  „         hundredth      „ 

and  so  on,  give  nearer  and  nearer  approximations  to  the  velocity  at 
the  beginning  of  the  first  second. 

The  whole  foundation  of  Newton's  differential  calculus  is,  in  fact, 
contained  in  the  simple  question,  '  What  is  the  rate  at  which  the 
space  described  by  a  moving  point  increases?'  i.e.  What  is  the 
velocity  of  the  moving  point?  Newton's  notation  for  t?ie  velocity, 
i.  e.  the  rate  at  which  s  increases,  or  the  fluxion  of  s,  is  s.  This 
notation  is  very  convenient,  as  it  saves  the  introduction  of  a  second 
letter. 

29.  'I'he  preceding  definition  of  velocity  is  equally  applicable 
whether  the  point  move  in  a  straight  or  a  curved  line;  but,  since, 
in  the  latter  case,  the  direction  of  motion  continually  changes,  the 
mere  amount  of  the  velocity  is  not  sufficient  completely  to  describe 
the  motion,  and  we  must  have  in  every  such  case  additional  data 
to  thoroughly  specify  the  motion. 

In  such  cases  as  this  the  method  most  commonly  employed, 
whether  we  deal  with  velocities,  or  (as  we  shall  do  farther  on)  with 
accelerations  and  forces,  consists  in  studying,  not  the  velocity,  accele- 
ration, or  -force,  dincily,  but  its  resolved  parts  parallel  to  any  three 
assumed  directions  at  right  angles  to  each  other.  Thus,  for  a  train 
moving  up  an  incline  in  a  N.E.  direction,  we  may  have  the  whole 
velocity  and  the  steepness  of  the  incline  given ;  or  we  may  express 
the  same  ideas  thus — the  train  is  moving  simultaneously  northward, 
eastward,  and  upward; — and  the  motion  as  to  amount  and  direction 
will  be  completely  known  if  we  know  separately  the  northward,  east- 
ward, and  upward  velocities — these  being  called  the  components  of 
the  whole  velocity  in  the  three  mutually  perpendicular  directions 
N.,  E.,  and  up. 

80,  A  velocity  in  any  direction  may  be  resolved  m,  ana  perpen- 
dicular to^  any  other  direction.  The  first  component  is  found  by 
multiplying  the  velocity  by  the  cosine  of  the  angle  between  the  two 


8  PRELIMINARY. 

directions ;  the  second  by  using  as  factor  the  sine  of  the  same  angle. 
Thus  a  point  moving  with  velocity  V  up  an  Inclined  Plane,  making 
an  angle  a  with  the  horizon,  has  a  vertical  velocity  Vsix\  a  and  a 
horizontal  velocity  Fcos  a. 

Or  it  may  be  resolved  into  components  in  any  three  rectangular 
directions,  each  component  being  found  by  multiplying  the  whole 
velocity  by  the  cosine  of  the  angle  between  its  direction  and  that  of 
the  component.  The  velocity  resolved  in  any  direction  is  the  sum 
of  the  resolved  parts  (in  that  direction)  of  the  three  rectangular  com- 
ponents of  the  whole  velocity.  And  if  we  consider  motion  in  one 
plane,  this  is  still  true,,  only  we  have  but  two  rectangular  com- 
ponents. 

31.  These  propositions  are  virtually  equivalent  to  the  following 
obvious  geometrical  construction  : — 

To  compound  any  two  velocities  as  OA,  OB  in  the  figure ;  where 
n  ^    OA,  for  instance,  represents  in  magni- 

/" "^    tude  and  direction   the   space  which 

/       ^^■^^y^       would  be  described  in  one  second  by 

/   ^y^  X  a  point  moving  with  the  first  of  the 

/ ^^,,y^     /  given  velocities — and  similarly  OB  for 

j/^  /  the  second;  from  ^  draw  AC  parallel 

.££_ /  and  equal  to  OB.    Join  OC:  then  OC 

^  ^  is  the  resultant  velocity  in  magnitude 

and  direction. 

OC  '\%  evidently  the  diagonal  of  the  parallelogram  two  of  whose 
sides  are  OA,  OB. 

Hence  the  resultant  of  any  two  velocities  as  OA,  AC,  in  the 
figure,  is  a  velocity  represented  by  the  third  side,  OC,  of  the  triangle 
OAC. 

Hence  if  a  point  have,  at  the  same  time,  velocities  represented  by 
OA,  A  C,  and  CO,  the  sides  of  a  triangle  taken  in  the  same  order^  it 
is  at  rest. 

Hence  the  resultant  of  velocities  represented  by  the  sides  of  aay 
closed  polygon  whatever,  whether  in  one  plane  or  not,  taken  all  in 
the  same  order,  is  zero. 

Hence  also  the  resultant  of  velocities  represented  by  all  the  sides 
of  a  polygon  but  one,  taken  in  order,  is  represented  by  that  one 
taken  in  the  opposite  directioa 

When  there  are  two  velocities,  or  three  velocities,  in  two  or  in 
three  rectangular  directions,  the  resultant  is  the  square  root  of  the 
sum  of  their  squares ;  and  the  cosines  of  its  inclination  to  the  given 
directions  are  the  ratios  of  the  components  to  the  resultant. 

32.  The  velocity  of  a  point  is  said  to  be  accelerated  or  retarded 
according  as  it  increases  or  diminishes,  but  the  word  acceleration  is 
generally  used  in  either  sense,  on  the  understanding  that  we  may 
regard  its  quantity  as  either  positive  or  negative  :  and  (§  34)  is 
farther  generalized  so  as  to  include  change  of  direction  as  well  as 
change  of  speed,    Acceleration  of  velocity  may  of  course  be  either 


KINEMATICS,  9 

uniform  or  variable.  It  is  said  to  be  uniform  when  the  point  receives 
equal  increments  of  velocity  in  equal  times,  and  is  then  measured  by 
the  actual  increase  of  velocity  per  unit  of  time.  If  we  choose  as  the 
unit  of  acceleration  that  which  adds  a  unit  of  velocity  per  unit  of 
time  to  the  velocity  of  a  point,  an  acceleration  measured  by  a  will 
add  a  units  of  velocity  in  unit  of  time — and,  therefore,  a  t  units  of 
velocity  in  /  units  of  time.  Hence  if  v  be  the  change  in  the  velocity 
during  the  interval  /, 

v-at,  or  a  =  -. 

33.  Acceleration  is  variable  when  the  point's  velocity  does  not 
receive  equal  increments  in  successive  equal  periods  of  time.  It  is 
then  measured  by  the  increment  of  velocity,  which  would  have -been 
generated  in  a  unit  of  time  had  the  acceleration  remained  throughout 
that  unit  the  same  as  at  its  commencement.  The  average  accelera- 
tion during  any  time  is  the  whole  velocity  gained  during  that  time; 
divided  by  the  time.  In  Newton's  notation  v  is  used  to  express  the 
acceleration  in  the  direction  of  motion  j  and,  if  z/  =  i  as  in  §  28,  WQ 
have  a  —  ii-s. 

34.  But  there  is  another  form  in  which  acceleration  may  manliest 
itself.  Even  if  a  point's  velocity  remain  unchanged,  yet  if  its  direc- 
tion of  motion  change,  the  resolved  parts  of  its  velocity  in  fixed 
directions  will,  in  general,  be  accelerated. 

Since  acceleration  is  merely  a  change  of  the  component  velocity 
iii  a  stated  direction,  it  is  evident  that  the  laws  of  composition  and 
resolution  of  accelerations  are  the  same  as  those  of  velocities. 

We  therefore  expand  \}!i%  definition  just  given,  thus: — Acceleration 
•is  the  rate  of  change  of  velocity  ivhether  that  change  take  place  in  the 
direction  of  tnotion  or  not. 

35.  What  is  meant  by  change  of  velocity  is  evident  from  §  31. 
For  if  a  velocity  OA  become  OC,  its  change  is  AC,  or  OB. 

Hence,  just  as  the  direction  of  motion  of  a  point  is  the  tangent  to 
its  path,  so  the  direction  of  acceleration  of  a  moving 
point  is  to  be  found  by  the  following  construction: — 

From  any  point  6>,draw  lines  OP,  OQ,  etc.,  repre- 
senting in  magnitude  and  direction  the  velocity  of  the 
moving  point  at  every  instant.  (Compare  §  49.)  Tiie 
points,  P,  Q,  etc.,  must  form  a  continuous  curve,  for 
(^  7)  OP  cannot  change  abruptly  in  direction.  Now 
if  ()  be  a  point  near  to  P,  OP  and  OQ  represent  two 
successive  values  of  the  velocity.  Hence  PQ  is  the 
whole  change  of  velocity  during  the  interval.  As  the 
interval  becomes  smaller,  the  direction  PQ  more  and  more  nearly 
becomes  the  tangent  at  P.  Hence  the  direction  of  acceleration  is 
that  of  the  tangent  to  the  curve  thus  described. 

The  magnitude  of  the  acceleration  is  the  rate  of  change  of  velocity, 
and  is  therefore  measured  by  the  velocity  oi  P  in  the  curve  PQ. 


Id  PRELIMINARY. 

38.  Let  a  point  describe  a  circle,  ABD,  radius  R,  with  Uniform 
velocity  V,  Then,  to  determine  the  direction  of  acceleration,  we 
must  draw,  as  below,  from  a  fixed  point  O,  lines  OP,  GQ,  eta, 
representing  the  velocity  at  A,  B,  etc.,  in  direction  arid  magnitude. 
Since  the  velocity  in  ABD  is  constant,  all  the  lines  OP,  OQ,  etc., 
a^  will  be  equal  (to  F),  and  there- 

fore PQS  is  a  circle  whose 
centre  is  O.  The  direction  of 
ft  .acceleration  at  A  is  parallel  to 
the  tangent  at  P,  that  is,  is  per- 
pendicular to  OP,  i.e.  to  Aa, 
and  is  therefore  that  of  the 
radius  AC. 

Now  P  describes  the  circle 
PQS,  while  A  describes. ABD. 
Hence  the  velocity  of  P  is  to 
that  of  .^  as  OP  to  CA,  i.e.  as  Vto  R;  and  is  therefore  equal  tq 

r'^''r> 

and  this  (§  35)  is  the  amount  of  the  acceleration  in  the  circular  path 
ABZ>. 

37.  The'  whole  acceleration,  in  any  direction  is  the  sum  of  the 
components  (in  that  direction)  of  the  accelerations  parallel  to  any 
three  rectangular  axes — each  component  acceleration  being  found 
by  the  same  rule  as  component  velocities  (§  34),  that  is,  by  multiply- 
ing by  the  cosine  of  the  angle  between  the  direction  of  the  accelera- 
tion and  the  line  along  which  it  is  to  be  resolved. 

38.  When  a  point  moves  in  a  curve  the  whole  acceleration  may 
be  resolved  into  two  parts,  one  in  the  direction  of  the  motion  and 
equal  to  the  acceleration  of  the  velocity;  the  other  towards  the 
centre  of  curvature  (perpendicular  therefore  to  the  direction  of  mo- 
tion), whose  magnitude  is  proportional  to  the  square  of  the  velocity 
and  also  to  the  curvature  of  the  path.  The  former  of  these  changes 
the  velocity,  the  other  affects  only  the  form  of  the  path,  or  the 
direction  of  motion.  Hence  if  a  moving  point  be  subject  to  an 
acceleration,  constant  or  not,  whose  direction  is  continually  perpen- 
dicular to  the  direction  of  motion,  the  velocity  will  not  be  altered- 
and  the  only  effect  of  the  acceleration  will  be' to  make  the  point 
move  in  a  curve  whose  curvature  is  proportional  to  the  acceleration 
at  each  instant,  and  inversely  as  the  square  of  the  velocity. 

89.  In  other  words,  if  a  point  move  in  a  curve,  whether  with  a 
uniform  or  a  varying  velocity,  its  change  of  direction  is  to  be  re- 
garded as  constituting  an  acceleration  towards  the  centre  of  curva-* 
ture,  equal  in  amount  to  the  square  of  the  velocity  divided  by  the 
radius  of  curvature.  The  whole  acceleration  will,  in  every  case,  be 
the  resultant  of  the  acceleration  thus  measuring  change  of  direction, 
and  the  acceleration  of  actual  velocity  along  the  curve 


KINEMATICS.  rt 

40.  If  for  any  case  of  motion  of  a  point  we  have  given  th$  whole 
velocity  and  its  direction,  or  simply  the  components  of  the  velocity 
in  three  rectangular  directions,  at  any  time,  or,  as  is  most  commonly 
the  case,  for  any  position  ;  the  determination  of  the  form  of  the  path 
described,  and  of  other  circumstances  of  the  motion,  is  a  question  of 
pure  nvathematics,  and  in  all  cases  is  capable  (if  not  of. an  exact 
solution,  at  all  events)  of  a  solution  to  any  degree  of  approximation 
that  may  be  desired 

This  is  true  also  if  the  total  acceleration  and  its  dh-ection  at  every 
instant,  or  simply  its  rectangular  components,  be  given,  provided  the 
velocity  and  its  direction,  as  well  as  the  position  of  the  point,  at  any 
one  instant  be  given.  But  these  are,  in  general,  questions  requiring 
for  their  solution  a  knowledge  of  the  integral  calculus. 

41.  From  the  principles  already  laid  down,  a  great  many  interest- 
ing results  may  be  deduced,  of  which  we  enunciate  a  few  of  the 
simpler  and  more  important. 

(a)  If  the  velocity  of  a  moving  point  be  uniform,  and  if  its  direction 
revolve  uniformly  in  a  plane,  the  path  described  is  a  circle. 

{b)  If  a  point  moves  in  a  plane,  and  its  component  velocity 
parallel  to  each  of  two  rectangular  axes  is  proportional  to  its  dis- 
tance from  that  axis,  the  path  is  an  ellipse  or  hyperbola  whose 
{Principal  diameters  coincide  with  those  axes ;  and  the  acceleration 
is  directed  to  or  from  the  centre  of  the  curve  at  every  instant 
(§§  66,  78). 

{c)  If  the  comp6nents  of  the  velocity  parallel  to  each  axis  be  equi- 
multiples of  the  distances  from  the  other  axis,  the  path  is  a  straight 
line  passing  through  the  origin. 

{d)  When  the  velocity  is  uniform,  but  in  a  direction  revolving 
uniformly  in  a  right  circular  cone,  the  motion  of  the  point  is  in  a 
circular  helix  whose  axis  is  parallel  to  that  of  the  cone. 

42.  When  a  point  moves  uniformly  in  a  circle  of  radius  R,  with 
velocity  V,  the  whole  acceleration  is  directed  towards  the  centre,  and 

V 

has  the  constant  value  --5-.    See  §  36, 

43.  With  uniform  acceleration  in  the  direction  of  motion,  a  pofnt 
describes  spaces  proportional  to  the  squares  of  the  times  elapsed 
since  the  commencement  of  the  motion.  This  is  the  case  of  a  body 
falling  vertically  in  vacuo  under  the  action  of  gravity. 

Ift  this  case  the  space  described  in  any  interval  is  that  which  would 
\>t  described  in  the  same  time  by  a  point  moving  uniformly  with  a 
velocity  equal  to  that  at  the  middle  of  the  interval.  In  other  words, 
the  average  velocity  ^when  the  acceleration  is  uniform)  is,  during  any 
interval,  the  arithmetical  mean  of  the  initial  and  final  velocities.  For, 
since  the  velocity  increases  uniformly,  its  value  at  any  time  before  the 
middle  of  the  interval  is  as  much  less  than  this  mean  as  its  value 
at  the  same  time  after  the  middle  of  the  interval  is  greater  than  the. 


12  I'Reliminary: 

fnean :  and  hence  hs  value  at  the  middle  of  the  interval  must  be  the 
mean  of  its  first  and  last  values. 

In  symbols;  if  at  time  /=o  the  velocity  was  Vy  then  at  time  /  it  is 
v=  V-hat. 
Also  the  space  {x)  described  is  equal  to  the  product  of  the  time  by  the 
average  velocity.    But  we  have  just  shown  that  the  average  velocity  is 

and  therefore  a:  4  ^.  +  ^af'. 

Hence,  by  algebra, 

F'  +2ax  =  F'+2  Vat.+  (ff  =  {V^  atj  =  ^, 
or  ^  V* -  \V^ ti ax^ 

If  there  be  no  initial  velocity  our  equations  become 

v^atj    X  =  ^af,    ^j/  =  ax. 
Of  course  the  preceding  formulae  apply  to  a  constant  retardation,  as 
in  the  case  of  a  projectile  moving  vertically  upwards,  by  simply  giving 
a  a  negative  sign. 

4tfk.  When  there  is  uniform  acceleration  in  a  constant  direction, 
the  path  described  is  a  parabola,  whose  axis  is  parallel  to  that 
direction.     This  is  th^  case  of  a  projectile  moving  in  vacuo. 

For  the  velocity '( V)  in  the  original  direction  of  motion  remams 
unchanged ;  and  therefore,  in  time  /,  a  space  Vt  is  described  pafallel 
to  this  line.  But  in  the  same  interval,  by  the  above  reasoning,  we  see 
that  a  space  laf  is  described  parallel  to  the  direction  of  acceleration. 
Hence,  \i  AF  be  the  direction  of  motion  at  A,  AB  the  direction 
of  acceleration,"  and  Q  the  position  of  the  point  at-  time./j 
draw  QP  parallel  to  £A,  meeting  AF  m, 
F:  then 

AF^Vt,    FQ=^^at\ 

Hence 

zV* 

a       ^ 
This  is  a  property  of  a  parabola,  of  which 
the  axis  is  parallel  to  AB',  AB  being  a 
diameter,  and  AF  a  tangent.      If  O  be 
the  focus  of  this  curve,  we  know  that 
AP'  =  aOA.FQ, 

Hence 

F* 

i»  2a 

and  is  therefore  known.  Also  OA  is  known  in  direction,  for  AP 
bisects  the  angle,  OAC,  between  the  focal  distance  of  a  point  and 
the  diameter  through  it. 

45.     When  the  acceleration,  whatever  (and  however  varying)  ber 
its  magnitude,  is  directed  to  a  fixed  point,  the  path  is  in  a  plane 


KINEMATICS.  13 

passing  througji  that  point ;  and  in  this  plane  the  areas  traced  out  by 

the  radius-vector  are  proportional  to  the  times  employed. 

Evidently  ther^  is  no  acceleration  perpendicular  to  the  plane  con- 
taining the  tixed  point  and  the  line  of  motion  of  the  moving  point 
at  any  instant;  and  there  being  no  velocity  perpendicular  to  this 
plane  at  starting,  there  is  therefore  none  throughout  the  motion ; 
thus  the  point*^noves  in  the  plane.  For  the  proof  of  the  second 
part  of  the  proposition  we  must  make  a  slight  digression.. 

.  46.  The  Moment  of  a  velocity  or  of  a  force  about  any  point  is 
the  product  of  its  magnitude  into  the  perpendicular  from  the  point 
upon  its  direction.  The  moment  of  the  resultant  velocity  of  a  par- 
ticle about  any  point  in  the  plane  of  the  components  is  equal  to  the 
algebraic  sum  of  the  moments  of  the  components,  the  proper  sign  of 
each  moment  depending  on  the  direction  of  motion  about  the  point. 
The  same  is  true  of  moments  of  forces  and  of  moments  of  momentum, 
as  defined  in  Chapter  11. 

First,  consider  two  component  motions,  AB  and  A  C,  and  let  AD 
be  their  resultant  (§  31).  Their  half-moments  round  the  point  O 
are  respectively  the  areas  OAB,  OCA.  Now  OCA,  together  with 
half  the  area  of  the  parallelogram  CABD,  is  equal  to  OBD.  Hence 
the  sum  of  the  two   half-moments  together  x% 

with   half  the   area  of  the  parallelogram  is  /:\-.._ 

equal  to  AOB  together  with  BOD,  that  is  ■\'''\ 

to   say,    to   the    area   of   the   whole   figure  /  •    \    ""v 

OABD.  But  ABD,  a  part  of  this  figure, 
is  equal  to  half  the  area  of  the  parallelo- 
gram ;  and  therefore  the  remainder,   OAD, 

is  equal   to   the  sum  of  the   two    half-mo-     

ments.     But  OAD  is  half  the  moment  of  the  4  U 

resultant  velocity  round  the  point  O.  Hence  the  moment  of  the 
resultant  is  equal  to  the  sum  of  the  moments  of  the  two  components. 
By  attending  to  the  signs  of  the  moments,  we  see  that  the  proposi- 
tion holds  when  0  is  \\ithin  the  angle  CAB. 

If  there^be  any  number  of  component  rectilineal  motions,  we  may 
compound  them  in  order,  any  two  taken  together  first,  then  a  third, 
and  soon;  and  it  followsthat  the  sum  of  their  moments  is  equal  to 
the  moment  of  their  resultant.  It  follows,  of  course,  that;  the  sum  of 
the  moments  of  any  number  of  component  velocities,  all  in  one  plane, 
into  which  the  velocity  of  any  point  may  be  resolved,  is  equal  to  the 
moment  of  their  resultant,  round  any  point  in  their  plane.  It  follows 
also,  that  if  velocities,  in  diftcrent  directions  all  in  one  plane,  be  suc- 
cessively given  to  a  moving  point,  so  that  at  any  time  its  velocity  is 
their  resultant,  the  moment  of  its  velocity  at  any  time  is  tlie  sum  of  the 
moments  of  all  the  velocities  which  have  been  successively  given  to  it. 

47.  Thus  if  one  of  the  components  always  passes  through  the 
pomt,  Its  moment  vanishes.  This  is  tlie  case  of  a  motion  in  which 
the  acceleration  is  directed  to  a  fixed  point,  and  we  thus  prove  the 
second  theorem  of  §  45,  that  in  the  case  supposed  the  areas  described 


14  PRELIMINARY. 

by  the  radius-vector  are  proportional  to  the  times;  for,  as  we  have 
seen,  the  moment  of  the  velocity  is  doable  the  area  traced  out  by  tha 
radius-vector  in  unit  of  time. 

48.  Hpnce  in.  this  case  the  velocity  at  any  point  is  inversely  as 
the  perpendicular  from  the  fixed  point  to  the  tangent  to  the  path  or 
the  momentary  direction  of  motion. 

For  the  product  of  this  perpendicular  and  the  velocity  at  any 
instant  gives  double  the  area  described  in  one  second  about  the 
■fixed  point,  which  has  just  been  shown  to  be  a  constant  quantity. 

As  the  kmematical  propositions  with  which  we  are.  deaUng  have 
important  bearings  on  Physical  Astronomy,  we  enunciate  here  Keple/s 
'Jjiws  of  Planetary  Motion.  They  were  deduced  originally  from 
observation  alone,  but  Newton  explained  th^m  on  physical  principles 
and  showed  that  they  are  applicable  to  comets  as  well  as  to  planets. 

I.  Each  planet  describes  an  Ellipse  [with  comets  this  may  be  any 
Conic  Section]  of  which  the  Sun  occupies  one  focus. 

II.  The  radius-vector  of  each  planet  describes  equal  areas. ia 
equal  times. 

III.  The  square  of  the  periodic  time  [in  an  elhptic  orbit]  is  pro* 
portional  to  the  cube  of  the  major  axis. 

Sections  45 — 47,  taken  in  connexion  with  the  second  of  these 
laws,  show  that  the  acceleration  .of  the  motion  of  a  planet  or  Comet 
is  along  the  radius-vector. 

49.  If,  as  in  §  35,  from  any  fixed  point,  lines  be  drawn  at  every 
instant  representing  in  magnitude  and  direction  the  velocity  of  a  point 
describing  any-  path  in  any  manner,  the  extremities  of  these  lines 
form  a  curve  which  is  called  the  Hodograph.  The  fixed  point  fi-om 
which  these  lines  are  drawn  is  called  the  hodographic  origin.  The 
invention  of  this  construction  is  due  to  Sir  W.  R.  Hamilton;  and; 
one  of  the  most  beautiful  of  the  many  remarkable  theorems  to  which 
it  leads  is  this  :  The  Hodograph  for  the  motion  of  a  planet  or  comet  it, 
always  a  circle,  whatever  be  the  form  and  dimensions  of  the  orbit.  The 
proof  will  be  given  immediately. 

It  was  shown  (§  35)  that  an  arc  of  the  hodograph  represents  the 
change  of  velocity  of  the  moving  point  during  the  corresponding 
time;  and  also  that  the  tangent  to  tlae  hodograph  is  parallel  to  the 
direction,  and  the  velocity  in  the  hodograph  is  equal  to  the  amount 
of  the  acceleration  of  the  moving  point 

When  the  hodograph  and  its  origin,  and  the  velocity  along  it,  or 
the  time  corresponding  to  each  point  of  it,  are  given,  the  orbit  may 
easily  be  shown  to  be  determinate. 

[An  important  improvement  in  nautical  charts  has  been  suggested 
by  Archibald  Smith'.  It  consists  in  drawing  a  curve,  which  may 
be  called  the  ridal  hodograph  with  reference  to  any  point  of  a  chart 
for  which  the  tidal  currents  are  to  be  specified  throughout  the  chief 
tidal  period  (twelve  lunar  hours).  Numbers  from  I.  to  XII.  are  placed 
at  marked  points  along  the  curve,  corresponding  to  the  lunar  hourse 

«  Proc.  R.  S.  1865. 


KINEMATICS. 


15 


"Smith's  curve  is  precisely  the  Hamiltonian  hodograph  for  an  imagi- 
nary particle  moving  at  each  instant  with  the  same  velocity  and  the 
same  direction  as  the  particle  of  fluid  passing,  at  the  same  instant, 
through  the  point  referred  to.] 

50.  In  the  case  of  a  projectile  (§  44),  the  horizontal  component 
of  the  velocity  is  unchanged,  and  the  vertical  component  increases 
uniformly.  Hence  the  hodograph  is  a  vertical  straight  line,  whose 
distance  from  the  origin  is  the  horizontal  velocity,  and  which  is 
described  uniformly. 

51.  To  prove  Hamilton's  proposition  (§  49),  let  APB  be  a  portion 
of  a  conic  section  and  S  one  focus.  Let  P  move  so  that  SP 
describes  equal  areas  in  equal  times,  that  is  (§  48), 
let  the  velocity  be'  inversely  as  the  perpendicular 
SY  from  vS  to  the  tangent  to  the  orbit.  \i  ABP 
be  an  ellipse  or  hyperbola,  the  intersection  of  the 
perpendicular  with  the  tangent  lies  in  the  circle 
YAZ,  whose  diameter  is  the  niJljor  axis.  Produce 
YS  to  cut  the  circle  again  in  Z.  Then  YS.  SZ  is 
constant,  and  therefore  SZ  is  inversely  as  SY,  that 
is,  SZ  is  proportional  to  the  velocity  at  P.  Also 
SZ  is  perpendicular  to  the  direction  of  motion  PY, 
and  thus  the  circular  locus  of  Z  is  the  hodograph  turned  through  a 
right  angle  about  .S  in  the  plane  of  the  orbit.  If  APB  be  a  parabola, 
AY  IS  3i  straight  line.  But  if  another  point  ^be  taken  in  YS  pro- 
duced, so  that  YS.SUii  constant,  the  locus  of  6''is  easily  seen  to  be  a 
circle.  Hence  the  proposition  is  generally  true  for  all  conic  sections. 
The  hodograph  surrounds  its  origin  [as  the  figure  shows]  if  the  orbit 
be  an  ellipse,  passes  through  it  when  the  orbit  is  a  parabola,  and  the 
origin  is  without  the  hodograph  if  the  orbit  is  a  hyperbola. 

52.  A  reversal  of  the  demonstration  of  §  51  shows  that,  if  the 
acceleration  be  towards  a  fixed  point,  and  if  the  hodograph,  be  a 
circle,  the  orbit  must  be  a  conic  section  of  which  the  fixe"ci  point 
is  a  focus. 

-  But  we  may  also  prove  this  important  proposition  as  follows ; 
Let  A  be  the  centre  of  the  circle,  and  O  the  hodographic  origin. 
Join  OA  and  draw  the  perpendiculars 
PM  to  OA  and  ON  to  PA.  Then  OP 
is  the  velocity  in  the  orbit :  and  ON,  being 
parallel  to  the  tangent  at  P,  is  the  direc- 
tion of  acceleration  in  the  orbit ;  and  is 
therefore  parallel  to  the  radius-vector  to 
the  fixed  point  about  which  there  is  equable 
description  of  areas.  The  velocity  parallel 
to  the  radius-vector  is  therefore  ON,  and 
the  velocity  perpendicular  to  the  fixed  line 


OA  is  PM. 


But^^^=^-^^ 

PM     AP 


■■  constant. 


i6  PRELIMINARY. 

Hence,  in  the  orbit,  the  velocity  along  the  radius-vector  is  pro- 
portional to  that  perpendicular  to  a  fixed  line :  and  therefore  the 
radius-vector  of  any  point  is  proportional  to  the  distance  of  that 
point  from  a  fixed  line — a  property  belonging  exclusively  to  the 
conic  sections  referred  to  their  focus  and  directrix. 

53.  The  path  which,  in  consequence  of  Aberration^  a  fixed  star 
seems  to  describe,  is  the  hodograph  of  the  earth's  orbit,  and  is 
therefore  a  circle  whose  plane  is  parallel  to  the  plane  of  the 
ecliptic. 

54.  When  a  point  moves  in  any  manner,  the  line  joining  it  with 
a  fixed  point  generally  changes  its  direction.  If,  for  simplicity,  we 
consider  the  motion  to  be  confined  to  a  plane  passing  through  the 
fixed  point,  the  angle  which  the  joining  line  makes  with  a  fixed  line 
in  the  plane  is  continually  altering,  and  its  •  rate  of  alteration  at  any 
■in-stant  is  called  the  Angular  Velocity  of  the  first  point  about  the 
second.  .If  uniform,  it  is  of  course  measured  by  the  angle  described 
in  unit  of  time ;  if  variable,  by  the  angle  Avhich  would  have  been 
described  in  unit  of  time  if  the  angular  velocity  at  the  instant  in. 
question  were  maintained  constant  for  so  long.  In  this  respect  the 
process  is  precisely  similar  to  that  which  we  have  already  explained 
for  the  measurement  of  velocity  and  acceleration. 

We  may  also  speak  of  the  angular  velocity  of  a  moving  plane 
with  respect  to  a  fixed  one,  as  the  rate  of  increase  of  the  angle 
contained  by  them ;  but  unless  their  line  of  intersection  remain  fixed, 
or  at  all  events  parallel  to  itself,  a  somewhat  more  laboured  statement 
is  required  to  give  a  complete  specification  of  the  motion, 

55.  The  unit  angular  velocity  is  that  of  a  point  which  describes,^ 
or  would  describe,  unit  angle  about  a  fixed  point  in  unit  of  time.  The 
usual-unit  angle  is  (as  explaiiAed  in  treatises  on  plane  trigonometry)  that 
which  subtends  at  the  centre  of  a  circle  an  arc  whose  length  is  equal. 

to  the  radius;  being  an  angle  of =57°*29578  ...  =  57''i7'44""8 

nearly. 

58.  The  angular  velocity  of  a  point  in  a  plane  is  evidently  to  be 
found  by  dividing  the  velocity  perpendicular  to  the  radius-vector  by 
the  length  of  the  radius-vector. 

57.  When  the  angular  velocity  is  variable  its  rate  of  increase  or 
diminution  is  called  the  Angular  Acceleration^  and  is  measured  witli 
reference  to  the  same  unit  angle. 

^8.  When*  one  point  describes  uniformly  a'  circle  about  another, 
the  time  of  describing  a  complete  circumference  being  T,  we  have 
the  angle  z-ir  described  uniformly  ia.  T;  and,  therefore,  the  angqjar 

velocity  is  -;= .    Even  wher^.  the  angular  velocity  is  not  uniform,  as  in 

a-  planetfs  motion^  it  is  useful  to  introduce,  the  quantity  -y; ,  which  is 
then  called  the  mean  angular  velocity. 


KINEMATICS..  17 

59.  When  a  point  moves  unifomily  in  a  straight  line  its  angular 
velocit}'  evidently  diminishes  as  it  recedes  from  the  point  about 
■which  the  angles  are  measured,  and  it  may  easily  be  showTi  that 
it  varies  inversely  as  the  square  of  the  distance  from  this  point. 
The  same  proposition  is  true  for  any  path,  when  the  acceleration  is 
towards  the  point  about  which  the  angles  are  measured :  being 
inerely  a  different  mode  of  stating  the  result  of  §  48. 

60.  The  intensity  of  heat  and  light  emanating  from  a  point,  or. 
from  a  uniformly  radiating  spherical  surface,  diminishes  according  to 
ihe  inverse  square  of  the  distance  from  the  centre.  Hence  the  rate 
at  which  a  planet  receives  heat  and  light  from  the  sun  varies  in 
simple  proportion  to  the  angular  velocity  of  the  radius-vector.  Hence 
the  whole  heat  and  light  received  by  the  planet  in  any  time  is  pro- 
portional to  the  v.'hole  angle  turned  through  by  its  radius-vector  in_ 
the  same  time. 

6L  A  further  instance  of  this  use  of  the  idea  of  angular  velocity 
may  now  be  given,  to  sohe  the  problem  of  finding  the  hodograph. 
(§  35)  for  any  case  of  motion  in  which  the  acceleration  is  directed  to 
.a  fixed  point,  and  varies  inversely  as  the  square  of  the  distance  Irom 
-that  point.  Tiie  velocity  of  F,  in  the  holograph  PQ,  being  the 
acceleration  in  the  orbit, .  varies  inversely  as  the  square  of  the 
radius-vector;  and  therefore  (§  50)  directly  as  the 
angular  velocity.  Hence  the  arc  of  PQ,  described 
in  any  time,  is  proportional  to  the  corresponding 
angle-vector  in  the  orbit,  i.e.- to  the  angle  through 
which  the  tangent  to  PQ  has  turned.  Hence  (5  y) 
the  curvature  of  PQ  is  constant,  or  PQ  is  a  circle. 

This  demonstration,   reversed,    proves   that   if  the 
hodograph  be   a  circle,  and   the  acceleration  be  to- 
wards a  fixed  point,  the  acceleration  varies  inversely 
as  the  square  of  the  distance  of  the   moving  point      O 
from  tb.e  fixed  point. 

G2.  From  S;  61,  52,  it  follows  that  v,-hen  a  particle  moves  witii 
acceleration  towards  a  fixed  po'r.t,  \-arying.  inversely  as  the  squ.ire 
of  the  distance,  its  orbit  is  a  conic  section,  with  this  point  for  one 
focus.  And  conversely  (>.$  .;;,  51,  52),  if  the  orbit  be  a  conic  sec- 
tion, the  acceleration,  if  towards  either  focus,  varies  in\ersely  as  the 
•square  of  the  distance  :  or,  if  a  jioint  nio\e3  in  a  conic  section, 
■describing  equal  areas  in  equal  times  Ijv  a  radius-vector  thii'OiiL'h 
a  focus,  the  acceleratior.  is  alv.ays  tov\'ards  this  focus,  and  ^•ar;es 
inversely  as  the  scjuare  of  the  distance.  Curnpare  this  with  the  first 
and  Second  of  Keider's  La.w3,  §  aS. 

63.  All  motioit  that  v.e  are,  or  can  be,  acquainted  with,  \<.Rc'ai::e 
merely.  We  can  calculate  from  astronomical  data  for  any  instant 
.the  direction  in  which,  and  the  velocity  v.'ith  which,  we  are  movin:,^ 
on  account  of  the  earth's  diurnal  rotation.  We  may  compound  th.is 
with  the  (equally  calculable)  velocity  of  the  earth  in  its  orbit.  Tiiis 
•jesultant  again  wc  may  compound  with  the  (roughly-known)  %e:ocity 

Vui.  -n—i 


i8  PRELIMINARY. 

of  the  sun  relatively  to  the  so-called  fixed  stars;  but,  even  if  all 
these  elements  were  accurately  known,  it  could  not  be  said  that  we 
had  attained  any  idea  of  an  absolute  velocity ;  for  it  is  only  the  sun's 
relative  motion  among  the  stars  that  we  can  observe ;  and,  in  all 
probability,  sun  and  stars  are  moving  on  (it  may  be  with  incon- 
ceivable rapidity)  relatively  to  other  bodies  in  space.  We  must- there- 
fore consider  how,  from  the  actual  motions  of  a  set  of  bodies,  we 
may  find  their  relative  motions  with  regard  to  any  one  of  them ;  and 
how,  having  given  the  relative  motions  of  all  but  one  with  regard  to 
the  latter,  and  the  actual  motion  of  the  latter,  we  may  find  the  actual 
motions  of  all.  The  question  is  very  easily  answered.  Consider 
for  a  moment  a  number  of  passengers  walking  on  the  deck  of  a 
steamer.  Their  relative  motions  with  regard  to  the  deck  are  what  we 
immediately  observe,  but  if  'we  compound  with  these  the  velocity  of 
the  steamer  itself  we  get  evidently  their  actual  motion  relatively  to 
the  earth.  Again,  in  order  to  get  the  relative  motion  of  all  with 
regard  to  the  deck,  we  eliminate  the  motion  of  the  steamer  alto- 
gether; that  is,  we  alter  the  velocity  of  each  relatively  to  the  earth 
by  compounding  with  it  the  actual  velocity  of  the  vessel  taken  in 
a  reversed  direction. 

Hence  to  find  the  relative  motions  of  any  set  of  bodies  with  regard 
to  one  of  their  number,  imagine,  impressed  upon  each  in  composition- 
with  its  own  motion,  a  motion  equal  and  opposite  to  the  motion  of 
that  one,  which  will  thus  be  reduced  to  rest,  while  the  motions  of  the 
others  will  remain  the  same  with  regard  to  it  as  before. 

Thus,  to  take  a  very  simple  example,  two  trains  are  running  irv 
jjpposite  directions,  say  north  and  south,  one  with  a  velocity  of  fifty, 
the  other  of  thirty,  miles  an  hour.  The  relative  velocity  of  the  second 
with  regard  to  the  first  is  to  be  found  by  imagining  impressed  on 
both  a  southward  velocity  of  fifty  miles  an  hour ;  the  effect  of  this 
being  to  bring  the  first  to  rest,  and  to  give  the  second  a  southward 
velocity  of  eighty  miles  an  hour,  which  is  the  required  relative 
mofion. 

Or,  given  one  train  moving  north  at  the  rate  of  thirty  miles  an 
hour,  and  another  moving  west  at  the  rate  of  forty  miles  an  hour. 
The  motion  of  the  second  relatively  to  the  first  is  at  the  mte  of  fifty 
miles  an  hour,  in  a  south-westerly  direction  inclined  to  the  due  west 
direction  at  an  angle  of  tan"^  -f .  It  is  needless  ■  to  multiply  such- 
examples,  as  they  must  occur  to  every  one. 

64.  Exactly  the  same  remarks  apply  to  relative  as  compared  with 
absolute  acceleration,  as  indeed  we  may  see  at  once,  since  accelera-- 
tions  are  in  all  cases  resolved  and  compounded  by  the  same  law  a& 
velocities. 

65,  The  following  proposition  m  relative  motion  is  of  consider- 
able importance  :  — 

Any  two  moving  points  describe  similar  paths  relatively  to  each,' 
other  and  relatively  to  any  point  which  divides  in  a  constant  ratio 
the  line  joining  them. 


KINEMATICS.  19 

Let  A  and  B  be  zxij  simultaneous  positions  of  the  points.    Take 

G  or  G'  in  AB  such  that  the  ratio  -^-^  or  >,7-,,  has  a  constant 

value.     Then,  as  the  form  of  the  relative     t  ^  , 

path  depends  only  upon  the  length   and     (x        A.      G  B 

direction  of  the  line  joining  the  t^vo  points  at  any  instant,  it  is  obvious 
that  these  ■will  be  the  same  for  A  with  regard  to  B,  as  for  B  -with 
regard  to  A,  saving  only  the  inversion  of  the  direction  of  the  joining 
line.  Hence  Bs  path  about  A  \s  A's  about  B  turned  through  two 
right  angles.  And  with  regard  to  G  and  G'  it  is  evident  that  the 
directions  remain  the  same,  while  the  lengths  are  altered  in  a  given 
ratio ;  but  this  is  the  definition  of  similar  curves. 

66.  An  excellent  example  of  the  transformation  of  relative  into 
absolute  motion  is  afforded  by  the  family  of  Cycloids.  "We  shall  in 
a  future  section  consider  their  mechanical  description,  by  the  ro'/.v.'g- 
of  a  circle  on  a  fixed  straight  line  or  circle.  In  the  meantime,  we 
take  a  different  form  of  statement,  which  however  leads  to  precisely 
the  same  result. 

The  actual  path  of  a  point  which  revolves  uniformly  in  a  circle 
about  another  point — the  latter  moving  uniformly  in  a  straight  line 
or  circle  in  the  same  plane — belongs  to  the  family  of  Cycloids. 

67.  As  an  additional  illustration  of  this  part  of  our  subject,  i^-e 
may  define  as  follov.-s  : 

If  one  point  yl  executes  any  motion  whatever  with  reference  to 
a  second  point  B ;  \i  B  executes  any  other  motion  with  reference 
to  a  third  point  C;  and  so  on — the  first  is  said  to  execute,  with 
reference  to  the  last,  a  movement  which  is  the  resultant  of  these 
several  movemcms. 

The  relative  position,  velocity,  and  acceleration  are  in  such  a  case 
the  geometrical  resultants  of  the  various  components  combined 
according  to  preceding  rules. 

53.  The  foriOv-.-ing  practical  methods  of  effecting  such  a  com- 
bination in  the  simple  case  of  the  movements  of  tv,-o  points  are 
useful  in  scientific  illustrations  and  in  certain  mechanical  arrange- 
ments. Let  two  moving  poinis  be  joined  by  a  uniform  elastic  string; 
the  middle  point  of  this  strir.g  will  evidently  execute  a  movement 
which  is  half  the  resultant  of  tlie  motions  of  the  two  points.  But  for 
drawing,  or  engraving,  0:  for  other  mechanical  applications,  the 
following  method  is  preferab-c  : — 

Ci^and  ED  are  rods  of  ecral  length  moving  freely  round  a  pivot 

.at  P,  which  passes  thrcUi-Ji  jie  middle  point  E 

of  each— C^^,  AD,  EB,  n-I  yS\v  are  rods  of  V" 

half  the   lengih    of   tlic    two    former,    and    so  \ 

pivoted  to  them  as  to  fori;!   a   pa'r    of  equal   q \_ \r» 

^hombi  CD,  EE,  whose  angles  can  be  altered'      \  ^E 

at  will.     Whatever  niotions,  whether  in  a  plane,        \  ^. 

.or  in  space  of  three  dimensions,  be  given  to     •    \_„_^ \ 

A  and  B,  P  will  evidently  be  subjected  to  half       A  D 

their  resultant. 


aa  PRELIMINARY. 

69.  Amongst  the  most  important  classes  of  motions  which  we 
have  to  consider  in  Natural  Philosophy,  there  is  one,  namely,  Har*. 
monk  Motion,  which  is  of  such  immense  use,  not  only  in  ordinary 
kinetics,  but  in  the  theories  of  sound,  light,  heat,  etc.,  that  we  make 
no  apology  for  entering  here  into  some  little  detail  regarding  it. 

70.  Def.  When  a  point  Q  moves  uniformly  in  a  circle,  the  per- 
pendicular QP  drawn  from  its  position  at  any 
instant  to  a  fixed  diameter  AA'  of  the  circle, 
intersects  the  diameter  in- a  point  P,  whose 

Q  /       /->         \    position  changes  by  a  simple  harmonic  tnoiion. 

Thus,  if  a  planet  or  satellite,  or  one  of  the 
constituents  of  a  double  star,  be  supposed  to 
move  uniformly  in  a  circular  orbit  about  its 
primary,  and  be  viewed  from  a  very  distant 
position  in  the  plane  of  its  orbit,  it  will  appear 
to  move  backwards  and  forwards  in  a  straight 

line  with  a  simple  harmonic  motion.     This  is  nearly  the  case  with 

such  bodies  as  the  satellites  of  Jupiter  when  seen  from  the  earth. 
Physically,  the  interest  of  such  motions  consists  in  the  fact  of  their 

being  approximately  those   of  the  simplest  vibrations  of  sounding 

bodies  such  as  a  tuning-fork  or  pianoforte-wire ;  whence  their  name ;. 

and  of  the  various  media  in  which  waves  of  sound,  light,  heat,  etc., 

are  propagated. 

71.  The  Amplitude  of  a  simple  harmonic  motion  is  the  range  on 
one  side  or  the  other  of  the  middle  point  of  the  course,  i.  e.  OA  or 
OA'  in  the  figure. 

An  arc  of  the  circle  referred  to,  or  any  convenient  angular  reck- 
oning of  it,  measured  from  any  fixed  point  to  the  unifo'-mly  moving 
point  Q,  is  the  Afgumctit  of  the  harmonic  motion. 

[The  distance  of  a  point,  performing  a  simple  harmonic  motion, 
from  the  middle  of  its  course  or  range,  is  a  simple  harmonic  fimdioti 
(if  the  time;  that  is  to  say 

a  cos  (;;/  -  e), 
where  a,  n,  c  are  constants,  and  t  represents  time.     The  argument  of 
this  function  is  what  we  have  defined  as  the  ajgument  of  the  motion. 
In  the  formula  above,  the  argument  is  ?it  -  ^.] 

The  Epoch  in  a  simple  harmonic  motion  is  the  interval  of  time 
which  elapses  from  the  era  of  reckoning  till  the  moving  point  first 
comes  to  its  greatest  elongation  in  the  direction  reckoned  as  positive, 
from  its  mean  position  or  the  middle  of  its  range.  [In  the  formula 
above,  put  in  the  form 


^cos 


■•'('-^)' 


e . 


~  is  the  epoch.]    Epoch  in  angular  measure  is  the  angle  described 

on  the  circle  of  reference  in  the  period  of  time  defined  as  the  epoch. 
[In  the  formula,  e  is  the  epoch  in  angular  measure.] 
The  Period  oi  a.  simple  harmonic  motion  is  the  time  which  elapses 


KINEMATICS.  2t 

from  any  instant  tin!il  the  moving  point  again  moves  in  the  same 
direction  through  the  same  position,  and  is  evidently  the  time  of 
revolution  in  the  auxiliary  circle.     [In  the  formula  the  period  is 
27r-| 
n  '^ 

The  Phase  of  a  simple  harmonic  motion  at  any  instant  is  an 
expression  used  to  designate  the  part  of  its  whole  period  which  it 
has  reached.  It  is  borrowed  from  the  popular  expression  '  phases  of 
the  moon.'  Thus  for  Simple  Harmonic  Motion  we  may  call  the 
first  or  zero-phase  that  of  passing  through  the  middle  position  in  the 
positive  direction.  Then  follow  the.  successive  phases  quarter-period, 
half-period,  three-quarters-period,  and  complete  period  or  return  to 
zero-phase.  Sometimes  it  is  convenient  to  reckon  phase  by  a  number 
or  numerical  expression,  which  may  be  either  a  reckoning  of  angle  or 
a  reckoning  of  time,  or  a  fraction  or  multiple  of  the  period.  Thus' 
the  positive  maximum  phase  may  sometimes  be  called  the  90'^  phase 

or  the  phase  -,  or  the  three-hour  phase,  if  the  period  be  12  hours,  or 

the  quarter-period  phase.  Or,  again,  the  phase  of  half  way  down 
from  positive  maximum  may  be  described  as  the  120°  phase  or  the 

—  phase,  or  the  \  period  phase.     This  particular  way  of  specifying 

phase  is  simply  a  statement  of  the  argument  as  defined  above  and 
measured  from  the  point  corresponding  to  positive  motion  through 
the  middle  position. 

72.  Those  common  kinds  of  mechanism,  for  producing  rectilineal 
from  circular  motion,  or  vice  versa,  in  which  a  crank  moving  in 
a  circle  works  in  a  straight  slot  belonging  to  a  body  which  can 
only  move  in  a  straight  line,  fulfil  strictly  the  definition  of  a  simple 
harmonic  motion  m  the  part  of  which  the  motion  is  rectilineal,  if  I'le 
motion  of  the  rotating  part  is  uniform. 

The  motion  of  the  treadle  in  a-spinmng-whcel  approxunatcs  to 
"tl;e  same  condition  when  the  wheel  moves  uniformly;  the  ap].iroxi- 
mation  being  the  closer,  the  smaller  is  the  angular  motion  of  tlie 
treadle  and  of  the  connecting  string.'  It  is  also  approximated  to 
more  or  less  closely  in  the  motion  of  the  piston  of  a  steam-engine 
connected,  by  any  of  the  several  methods  in 
use,  with  the  crank,  provided  always  the  ro- 
tatory motion  of  the  crank  be  uniform. 

73.  The  velocity  of  a  point  executing  a 
simple  harmonic  motion  is  a  simple  harmonic 
function  of  the  time,  a  quarter  of  a  period 
earlier  in  phase  than  the  displacement,  and 
having  its  maximum  value  equal  to  the  ve- 
locity in  the  circular  motion  by  which  the  given 
function  is  defined.  -^' 

For,  in  the  fig.,  if  V  be  the  velocity  in  the  circle,  it  may  be 
represented   by   OQ  in  a  direction  perpencPcular  to  its  own,  and 


22 


PRELIMINARY, 


therefore  by  OP  and  FQ  in  directions  perpendicular  to  those  lines.^ 
That  is,  the  velocity  of  F  in  the  simple  harmonic  motion  is 
j>Q  y 

J—  V or -^ PQ.\  which,  when  P  passes  through  C,  becomes  V, 

74.    The  acceleration  of  a  point  executing  a  simple  harmonic 

.motion  is  at  any  time  simply  proportional  to  the  displacement  from 

the  middle  point,  but  in  opposite  direction,  or  always  towards  the 

middle  point.    Its  maximum  value  is  that  with  which  a  velocity 

equal  to  that  of  the  circular  motion  would  be  g,cquired  in  the  time 

in  which  an  arc  equal  to  the  radius  is  described. 

V* 
For  in  the  fig.,  the  acceleration  of  Q,  (by  §  36)  is  -^  along  QO. 

Supposing,  for  a  moment,  QO  to  represent  the  magnitude  of  this 

;acceleratio"n,  we  may  resolve  it  into  ^/*,  PO.     The  acceleration  of 

P  is  therefore:  represented  on  the  same  scale  by  PO.     Its  magnitude 

V*   PO      v 
is  therefore -^•^:y^=  -^-^ /'C?,  which  is  proportional  to  PO,  and 

has  at  A  its  maximum  value,  -jjTy)  ^^  acceleration  under  which  the 

velocity  V  would  be  acquired  in  the  time  -~  as  stated.    Thus  we 

have  in  simple  harmonic  motion 
Acceleration 


Q&'- 


4^ 


Displacement 

Where  T  is  the  time  of  describing  the  circle,  or  the  period  of  the 
harmonic  motion. 

75.  Any  two  simple  harmonic  motions  in  one  line,  and  of  one 
period,  give,  when  compounded,  a  single  simple  harmonic  motion ; 
of  the  same  period;  of  amplitude  equal 
to  the  diagonal  of  a  parallelogram  de- 
scribed on  lengths  equal  to  their  am- 
plitudes measured  on  lines  meeting  at 
an  angle  equal  to  their  difference  of 
epochs;  and  of  epoch  differing  from 
their  epochs  by  angles  equal  to  those 
which  this  diagonal  makes  with  the 
two  sides  of  the  parallelogram.  Let  P 
and  P  be  two  points  executing  simple 
harmonic  motions  of  one  period,  and  in 
one  line  B'BCAA'.  Let  Q  and  Q[  be  the 
uniformly  moving  points  in  the  relative 
circles.  On  CQ  and  CQ,  describe  a 
parallelogram  SQCQ;  and  through  ^S  draw  SR  perpendicular  to 
PA'  produced.  We  have  PR^  C/*  (being  projections  of  the  equal 
and  parallel  lines  Q'S,  CQ,  on  CR\  Hence  CR=CP-^CP'; 
and  therefore  the  point  R  executes  the  resultant  of  the  motions  P 
and  P\    But  CSy  the  diagonal  of  the  parallelogram,  is  constant 


KINEMATICS.  2^ 

(since  the  angular  velocities  of  CQ  and  CQ'  are  equal,  and  therefore 
the  angle  QCQ^  is  constant),  and  revolves  with  the  same  angular 
velocity  as  CQ  or  CQ';  and  therefore  the  resultant  motion  is  simple 
harmonic,  of  amplitude  CS,  and  of  epoch  exceeding  that  of  the 
motion  of  /*,  and  falling  short  of  that  of  the  motion  of  F',  by  the 
angles  ^C^and  SCQ^  respectively. 

This  geometrical  construction  has  been  usefully  applied  by  the 
tidal  committee  of  the  British  Association  for  a  mechanical  tide- 
indicator  (compare  §  77  below).  An  arm  CQ'  turning  round  C 
carries  an  arm  Q'S  turning  round  Q'.  Toothed  wheels,  one  of  them 
fixed  with  its  axis  through  C,  and  the  others  pivoted  on  a  framework 
carried  by  CQ',  are  so  arranged  that  Q'S  turns  very  approximately  at 
the  rate  of  once  round  in  12  mean  lunar  hours,  i(  CQ'  be  turned  uni- 
formly at  the  rate  of  once  round  in  12  mean  solar  hours.  Days  and 
half-days  are  marked  by  a  counter  geared  to  .CQ'.  The  distance  of 
S  from  a  fixed  line  through  C  shows  the  deviation  from  mean  sea- 
level  due  to  the  sum  of  mean  solar  and  mean  lunar  tides  for  the  time 
of  day  and  year  marked  by  CQ'  and  the  counter. 

76.  The  construction  described  in  the  preceding  section  exliibits 
the  resultant  of  two  simple  harm'onic  motions,  whether  of  the  same 
period  or  not.  [If  they  are  very  neady,  but  not  exactly,  of  tlie  same 
period,  the  diagonal  of  the  parallelogram  will  not  be  constant,  but 
will  diminish  from  a  maximum  value,  the  sum  of  the  component 
amplitudes,  which  it  has  at  the  instant  when  the  phases  of  the 
component  motions  agree  ;  to  a^  minimum,  tl>e  difference  of  those 
amplitudes,  which  is  its  value  when  the  phases  differ  by  half  a  period. 
Its  direction,  which  always  must  be  nearer  to  the  greater  than  to  tiic 
less  of  the  two  radii  constituting  i.he  sides  of  the  parallelogram,  will 
oscillate  on  each  side  of  the  greater  radius  to  a  maximum  deviation 
amounting  on  either  side  to  the  angle  whose  sine  is  the  less  radius 
divided  by  the  greater,  and  reached  when  the  less  radius  de\  iate-^  by 
this  together  with  a  quarter  circumference,  from  the  greater.  The 
full  period  of  this  oscillation  is  the  time  in  which  either  radius  gams 
a  full  turn  on  the  other.  The  resultant  motion  is  therefore  not 
simple  harmonic,  but  is,  as  it  were,  simple  harmonic  with  periodi- 
cally increasing  and  diminishing  amplitude,  and  with  jjcriodieal 
acceleration  and  retardation  of  phase.  This  view  is  most  approjiriate 
for  the  case  in  which  the  periods  of  the  two  component  motions 
are  nearly  equal,  but  the  amplitude  of  one  of  tliem  much  greater  than 
that  of  the  other. 

To  fmd  the  amount  of  the  maximum  advance  and  maximum  back- 
wardness of  phase,  and  when  they  are  experienced,  let  CA  be  equal 
to  the  greater  half-amplitude.  From  A  as 
centre,  with  AB  the  less  half-amplitude  as  77^ 

radius,  describe  a  ciicle.     CB  touching  tliis 
circle  represents  the  most  deviated  resultant.  ^^^ [      \ 

Hence  CBA  is  a  right  angle  ;  and  C'^""'^ 

€mBCA^'^^. 


84  PRELIMINARY. 

The  angle  BCA  thus  found  is  the  amount  by  which  the  phase  of 
the  resultant  motion  is  advanced  or  retarded  relatively  to  that  pf  the 
larger  component;  and  the  supplement  of  BCA  is  the  difference  of 
phase  of  the  two  components  at  the  time  of  maximum  advance  or 
backwardness  of  the  resultant] 

77.  A  most  interesting  application  of  this  case  of  the  composition 
of  harmonic  motions  is  to  the  lunar  and  solar  tides ;  which,  except 
in  tidal  rivers,  or  long  channels  or  bays,  follow  each  very  nearly  the 
simple  harmonic  law,  and  produce,  as  the  actual  result,  a  variation 
of  level  equal  to  the  sum  of  variations  that  would  be  produced  by 
the  two  causes  separately. 

The  amount  of  the  lunar  tide  in  the  equilibriutn  theory  is  about 
2'i  times  that  of  the  solar.  Hence  the  spring  tides  of  this  theory 
are  3'i,  and  the  neap  tides  only  i"i,  each  reckoned  in  terms  of  the 
solar  tide;  and  at  spring-  and  neap  tides  the  hour  of  high  water  is 
that  of  the  lunar  tide  alone.  The  greatest  deviation  of  the  actual 
tide  from  the  phases  (high,  low,  or  mean  water)  of  the  lunar  tide 
alone,  is  about  "95  of  a  lunar  hour,  that  is,  "98  of  a  solar  hour  (being 
the  same  part  of  12  lunar  hours  that  28°  26',  or  the  angle  whose 

sine  is  — ,  is  of  -560°).     This  maximum  deviation  will  be  in  advance 
2-1  ^      ' 

or  in  arrear  according  as  the  crown  of  the  solar  tide  precedes  or 
follows  the  crowTi  of  the  lunar  tide;  and  it  will  be  exactly  reached 
when  the  interval  of  phase  between  the  two  component  tides  is  3*95 
Junar  hours.  That  is  to  say,  there  will  be  maximum  advance  of  the 
tim.e  of  high  water  approximately  \\  days  after,  and  maximura  retar- 
dation the  same  number  of  days  before,  spring  tides. 

78.  We  may  consider  next  the  case  of  equal  amp  itudes  in  the 
two  given  motions.  If  their  periods  are  equal,  their  resultant  is  a 
simple  harmonic  motion,  whose  phase  is  at  every  instant  the  mean 
of  their  phases,  and  whose  amplitude  is  equal  to  twice  the  ampli- 
tude, of  either  multiplied  by  the  cosine  of  half  the  difference  of 
their  phase.  The  resultant  is  of  course  nothing  when  their  phases 
difl'er  by  half  the  period,  and  is  a  motion  of  double  arnplilude  and 
of  phase  the  same  as  theirs  when  they  are  of  the  same  phase. 

When  their  periods  are  very  nearly,  but  not  quite,  equal  (their 
amplitudes  being  still  supposed  equal),  the  motion  passes  very  slowly 
from  the  former  (zero,  or  no  motion  at  all)  to  the  latter,  and  back, 
in  a  time  equal  to  that  in  which  the  faster  has  gone  once  oftener 
through  its  period  than  the  slower  has. 

In  practice  we  meet  with  many  excellent  examples  of  this  case, 
which  will,  however,  be  more  conveniently  treated  of  when  we  come 
to  apply  kinetic  principles  to  various  subjects  in  practical  mechanics, 
acoustics,  and  physical  optics ;  such  as  the  marching  of  troops  over  a 
suspension  bridge,  the  sympathy  of  pendulums  or  tuning-forts,  etc. 

70,  If  any  number  of  pulleys  be  so  placed  that  a  cord  passing 
from  a  fixed  point  half  round  each  of  them  has  its  iree  parts  all 
in  parallel  lines,  and  if  their  centres  be  moved  with  simple  harmonic 


KINEMATICS. 


z; 


motions  of  iny  ranges  and  any  periods  irr  line?  parallel  to  those 
lines,  the  unattached  end  of  the  cord  moves  with  a  complex  har- 
monic motion  equal  to  twice  the  sum  of  the  given  simple  har- 
monic motions.  This  is  the  principle  of  Sir  W.  Thomson's  tide- 
predicting  m.achine,  constructed  by  the  British  Association,  and  order- 
ed to  be  placed  in  South  Kensington  Museum,  availably  for  general 
use  in  calculating  beforehand  for  any  port  or  other  place  on  the  sea 
for  which  the  simple  harmonic  constituents  of  the  tide  ba\c  been  de- 
termined by  the  'harmonic  analysis'  applied  to  previous  obier\a- 
tions'.  We  may  exhibit,  graphically,  the  various  preceding  cases  of 
single  or  compound  simple  harmonic  motions  in  one  line  by  cur^-es 
in  which  the  abscissae  represent  intervals  of  time,  and  the  ordinates 
the  corresponding  distances  of  the  moving  point  from  its  mean 
position.  In  the  case  of  a  single  simple  harmonic  motion,  the 
corresponding  curve  would  be  that  described  by  the  point  P  in 
§  70,  if,  while  Q  maintained  its  uniform  circular  motion,  the  circle 
were  to  move  with  uniform  velocity  in  any  direction  perpendicular 
to  OA.  This  construction  gives  the  harmonic  curve,  or  curve  of 
sines,  in  v/hich  the  ordinates  are  proportional  to  tb.e  sir.es  of  the 
abscissae,  the  straight  line  in  which  0  moves  being  the  axis  of 
abscissae.  It  is  the  simplest  possible  form  assumed  by  a  vibrating 
string  ;  and  when  it  is  assumed  that  at  each  instant  the  motion 
of  every  particle  of  the  string  is  simple  harmonic.  When  the 
harmonic  motion  is  complex,  but  in  one  line,  as  is  the  case 
for  any  point  in  a  violin-,  harp-,  or  pianoforte-string  (diftering, 
as  these  do,  from  one  another  in  their  motions  on  account 
of  the  different  modes  of  excitation  used),  a  similar  construction 
may  be  made.  Investigation  regarding  complex  harmonic  functions 
has  led  to  results  of  the  highest  importance,  having  their  most 
general  expression  in  Fourier's  Theorem,  to  be  presently  enunciated. 
We  give  below  a  graphic  representation  of  the  composition  of  two 
simple  harmonic  motions  in  one  line,  of  equal  amj.'litudcs  and  of 
periods  which  are  as  i  :  2  and  as  2  :  3,  the  e[)Ocl:s  being  each 
a  quarter  circumference.  The  horizontal  line  is  the  axis  of  ab- 
scissae of  the  curves ;  the  vertical  line  to  the  left  of  each  being  the 
axis  of  ordinates.     In  the  first  case  the  slower  motion  goes  through 


'  See  British  Associ.-ition  Ti'inl  Committee's  Reports,   i8i58,   iSp,  i375:  or 
Lecture  on  Tides,  by  Sir  \V.  Thomson  (Collins,  GIa>g;o\v,  1S76). 


iS  PRELIMINARY, 

one  complete  period,  in  the  second  it  goes  through  two  period*. 
These  and  similaf  cases  when  the  periodic  times  are  not  coramen: 
surable,  will  be  again  treated  of  under  Acoustics. 

80.  We  have  next  to  consider  the  composition  of  simple  har- 
monic motions  in  different  directions.  In  th^  first  place,  we  see 
that  any  number  of  simple  harmonic  motions  of  one  period,  and 
of  the  same  phase,  superimposed,  produce  a  single  simple  harmonic 
motion  of  the  same  phase.  For,  the  displacement  at  any  instant- 
being,  according  to  the  principle  of  the  composition  of  motions,  the 
geometrical  resultant  of  the  displacements  due  to  the  component 
motions  separately,  these  component  displacements  in  the  case  sup- 
posed, all  vary  in  simple  proportion  to  one  another,  and  are  in 
constant  directions.  Hence  the  resultant  displacement  will  vary 
in  simple  proportion  to  each  of  them,  and  will  be  in  a  constant 
direction. 

JBut  if,  while  their  periods  are  the  same,  the  phases  of  the  several 
'component  motions  do  not  agree,  the  resultant  motion  will  generally 
be  elliptic,  with  equal  areas  described  in  equal  times  by  the  radius- 
vector  from  the  centre ;  although  in  particular  cases  it  may  be  uni- 
form circular,  or;  on  the  other  hand,  rectilineal  and  simple  harmonic 

81.  To  prove  this,  we  may  first  consider  the  case,  in  which  we 
have  two  equal  simple  harmonic  motions  given,  and  these  in  per- 
pendicular lines,  and  differing  in  phase  by  a  quarter  period.  Their 
resultant  is  a  uniform  circular  motion.  For,  let  BA,  BA'  be  their 
ranges ;  and  from  (?,  their  common  middle  point  as  centre,  describe 

.-;'  a  circle  through  A  A'  BB'.    The  given  motion 

--;7-^^,         of  P  in  BA  will  be  (§67)  defined  by  the 

/     ^'  NX         motion  of  a  point  (?  round  thi.:  circumference 

/  I  \\       of  this  circle  ;  and  the  same  point,  if  moving 

b\ — 7y — hA.A.     in  the  direction  indicated  by  the  arrov/,  will 

V  I"      •*   /        give   a   simple   harmonic  motion  of  i^,   in 

\         I       y  BA',  a  quartei  of  a  period  behind  that  of 

^-j.  .- ^  the  motion  of  P  in  BA.     But,  since  A'OA, 

-■-''  QPO,  and  QP 0  are  right  angles,  the  iigiire 

■QPOP  is  a  parallelogram,  and  therefore  Q  is  in  the  position  of  the 
displacement  compounded  of  OP  and  OP.  Hence  two  equal  simple 
unarmonic  motions  in  perpendicular  lines,  of  phases  differing  by  a 
quarter  period,  are  equivalent  to  a  uniform  circular  m.otion  of  radius 
■equal  to  the  maximum  displacement  of  either  singly,  and  in  the  direc- 
tion from  tiie  positive  end  of  the  range  of  the  component  iu  advance 
of  the  other  towards  the  positive  end  of  the  range  of  this  latter. 

82.  Now,  orthogonal  projections  of  simple  harmonic  motions  are 
clearly  simple  harmonic  with  unchanged  phase.  Hence,  if  we  pro- 
ject the  case  of  §  81  on  any  plane,  we  get  motion  in  an  ellipse,  of 
which  the  projections  of  the  two  component  ranges  are  conjugate 
diameters,  and  in  which  the  Yadius-vector  from  the  centre  describes 
«qual  areas  (being  the  projections  of  the  areas  described  by  the 
Tadius  of  the  circle)  in  equal  times.     But  the  plane  and  position  of 


KINEMATICS.  17 

the  circle  of  which  this  projection  is  taken  may  clearly  be  found  so 
as  to  fulfil  the  condition  of  having  the  projections  of  the  ranges 
coincident  with  any  two  given  mutually  bisecting  lines.  Hence  any 
two  given  simple  harmonic  motions,  equal  or  unequal  in  range,  and 
oblique  or  at  right  angles  to  one  another  in  direction,  provided  only 
they  differ  by  a  quarter  period  in  phase,  produce  elliptic  motion, 
ha\-ing  their  ranges  for  conjugate  axes,  and  describing,  by  the 
radius-vector  from  the  centre,  equal  areas  in  equal  times. 

83.  Returning  to  the  composition  of  any  number  of  equal  simpb 
harmonic  motions  in  lines  in  all  directions  and  of  all  phases  :  each 
component  simple  harmonic  motion  may  be  determinately  resolved 
into  two  in  the  same  line,  differing  in  phase  by  a  quarter  period, 
and  one  of  them  having  any  given  epoch.  We  may  therefore  reduce 
the  given  motions  to  two  sets,  differing  in  phase  by  a  quarter  period, 
those  of  one  set  agreeing  in  phase  with  any  one  of  the  given,  or 
with  any  other  simple  harmonic  motion  we  please  to  choose  (i.e. 
having  their  epoch  anything  we  please). 

All  of  each  set  may  (§  75)  be  compounded  into  one  simple  har- 
monic motion  of  the  same  phase,  of  determinate  amplitude,  in  a  de- 
terminate line;  and  thus  the  whole  system  is  reduced  to  two  simple 
fully-determined  harmonic  motions  differing  from  one  another  in 
phase  by  a  quarter  period. 

Now  the  resultant  of  two  simple  harmonic  motions,  one  a  quarter 
of  a  period  in  advance  of  the  other,  in  different  lines,  has  been 
proved  (§  82)  to  be  motion  in  an  ellipse  of  which  the  ranges  of  the 
component  motions  are  conjugate  axes,  and  in  which  equal  areas 
are  described  by  the  radius-vector  from  the  centre  in  equal  times. 
Hence  the  proposition  of  §  80. 

84.  We  must  next  take  the  case  of  the  composition  of  simple 
harmonic  motions  of  different  periods  and  in  different  lines.  In 
general,  whether  these  lines  be  in  one  plane  or  nOt,  the  line  of 
motion  returns  into  itself  if  the  periods  are  commensurable;  and  \i 
not,  not.     This  is  evident  "without  proof 

Also  we  see  generally  that  the  composition  of  any  nuinber  of 
simple  harmonic  morions  in  any  directions  and  of  any  periods,  may 
be  effected  by  adding  their  components  in  each  of  any  three  rect- 
angular directions.  The  final  resultant  motion  is  thus  fully  expressed 
by  formulae  giving  the  rectangular  co-ordinates  as  'complex  harmonic 
functions '  of  the  time. 

85.  By  far  the  most  interesting  case,  and  by  far  the  simplest, 
is  that  of  two  simple  harmonic  motions  of  any  periods,  whose 
directions  must  of  course  be  in  one  plane. 

Mechanical  methods  of  obtaining  such  combinations  will  be  after- 
wards described,  as  Veil  as  cases  of  their  occurrence  in  Optics  and 
Acoustics. 

We  may  suppose,  for  simplicit}',  the  t^vo  component  motions  to  tike 
place  in  perpendicular  directions.  Also,  it  is  easy  to  see  that  we  can 
only  have  a  reentering  curve  when  their  periods  are  commensurable. 


28 


PRELIMINARY, 


The  following  figures  represent  the  paths  -produced  by  the  com- 
bination of  simple  harmonic  motions  of  equal  amplitude  in  two  rect- 
angular directions,  the  periods  of  the  components  being  as  i  :  2, 
ajid  the  epochs  differing  successively  by  o,  i,  2,  etc.,  skteenths  of 
a  circumference. 


In  the  case  of  epochs  equal,  or  differing  by  a  multiple  of  tt,  the 
curve  is  a  portion  of  a  parabola,  and  is  gone  over  twice  in  opposite 
directions  by  the  moving  point  in  each  complete  period. 

If  the  periods  be  not  exactly  as  i  :  2  the  form  of  the  path  pro- 
duced by  the  combination  changes  gradually  from  one  to  another 
of  the  series  above  figured ;  and  goes  through  all  its  changes  in  the 
time  in  which  one  of  the  components  gains  a  complete  vibration  oa 
the  other. 

88.  Another  very  important  case  is  that  of  two  pairs  of  simple 
harmonic  motions  in  one  plane,  such  that  the  resultant  of  each  pair 
is  uniform  circular  motion. 

If  their  periods  are  equal,  we  have  a  case  belonging  to  those 
already  treated  (§  80),  and  conclude  that  the  resultant  is,  in  general/ 
motion  in  an  ellipse,  equal  areas  being  described  in  equal  times 
about  the  centre.  As  particular  c^ses  we  laay  have  simple  har- 
monic, or  uniform  circular,  motion. 

If  the  circular  motions  are  in  the  same  direction,  the  resultant  is 
evidently  circular  motion  in  the  same  direction.  This  is  the  case 
of  the  motion  of  5  in  §  75,  and  requires  no  further  comment,  as 
its  amplitude,  epoch,  etc.,  are  seen  at  once  from  the  figure. 


KINEMATICS. 


»9 


87.  If  the  radii  of  the  component  motions  are  equal,  and  the 
periods  very  nearly  equal,  but  the  motions  in  opposite  directions, 
•we  have  cases  of  great  import:ince  in  modern  physics,  one  of  which 
ds  figured  below  (in  general,  a  non-reentrant  curve). 


This  is  intimately  connected  with  the  explanation  of  two  sets  of 
important  phenomena, — the  rotation  of  the  plane  of  polarization  of 
light,  by  quartz  and  certain  fluids  on  the  one  hand,  and  by  trans- 
parent bodies  under  magnetic  forces  on  the  other.  It  is  a  case  of 
the  hypotrochoid,  and  its  corresponding  mode  of  description  will  be 
described  in  §  104.  It  may  be  exhibited  experimentally  as  the  path 
of  a  pendulum,  hung  so  as  to  be  free  to  move  in  any  vertical  phne 
through  its  point  of  suspension,  and  containing  in  its  bob  a  fly-wliccl 
in  rapid  rotation. 

88.  [Before  leaving  for  a  time  the  subject  of  the  composition  of 
harmonic  motions,  we  must  enunciate  Fourier's  Theorem,  which  is 
not  only  one  of  the  most  beautiful  results  of  modern  analysis,  but 
may  be  said  to  furnish  an  indispensable  irstrjment  in  tlic  t'-cAiinent 
■of  nearly  every  recondite  question  in  modern  physics.  To  mcntioii 
only  sonorous  vibrations,  the  propagation  of  electric  signals  alorig 
a  telegraph  wire,  and  the  conduction  of  heat  by  the  eartr."s  cru^t, 
as  subjects  in  their  generality  intractable  witb.out  it,  is  to  give  but 
a  feeble  idea  of  its  importance.  Unfortunately  it  is  impossible  to 
give  a  satisfactory  proof  of  it  without  introducing  some  ratb.cr  trouble- 
some analysis,  which  is  foreign  to  the  purpose  of  so  elementary  a 
treatise  as  the  present. 

The  fullov.ing  seems  to  be  the  most  intelligible  form  in  which  it 
can  be  presented  to  the  gen';ral  reader: — 

Theorem. — A  complex  harmonic  function,  "with  a  constant  th-ni 
added,  is  the  proper  expression,  in  matliematical  language,  for  any 
arbitrary  periodic  function  ;  and  consequently  can  express  any  function. 
ivlujtever  between  definite  values  of  the  variable. 


30  PRELIMINARY. 

89.  Any  arbitrary  periodic  function  whatever  being  given,  the 
amplitudes  and  epochs  of  the  terms  of  a  complex  harmonic  function, 
which  shall  be  equal  to  it  for  every  value  of  the  independent  variable, 
may  be  investigated  by  the  *  method  of  indeterminate  coefficients.' 
.Such  an  investigation  is  sufficient  as  a  solution  of  the  problem, — to 
find  a  complex  harmonic  function  expressing  a  giv^n  arbitrary 
periodic  function, — when  once  we  are  assured  that  the  problem  is 
possible;  and  when  we  have  this  assurance,  it  proves  that  the  reso* 
lution  is  determinate ;  that  is  to  say,  that  no  other  complex  harmonic 
function  than  the  one  we  have  found  can  satisfy  the  conditions.] 

90.  We  now  pass  to  the  consideration  of  the  displacement  of  a 
rigid  body  or  group  of  points  whose  relative  positions  are  unalterable. 
The  simplest  case  we  can  consider  is  that  of  the  motion  of  a  plane 
figure  in  its  own  plane,  and  this,  as  far  as  kinematics  is  concerned, 
is  entirely  summed  tip  in  the  result  of  the  next  section. 

91.  If  a  plane  figure  be  displaced  in  any  way  in  its  own  plane,, 
there  is  always  (with  an  exception  treated  in  §  93)  one  point  of  it 
common  to  any  two  positions  j  that  is,  it  may  be  moved  from  any 
one  position  to  any  other  by  rotation  in  its  own  plane  about  one 
point  held  fixed. 

To  prove  this,  let  A,  B  be  any  two  points  of  the  plane  figure  in  a 
.first  position,  A',  B'  the  position  of  the  same  two  after  a  displacement. 
The  lines  .AA\  BB'  \yill  not  be  parallel,  except  in  one  case  to  be 
presently  considered.  Hence  the  line  equidistant  from  A  and  A' 
will  meet  that  equidistant  from  B  and  B'  in  some  point  O.  Join 
OA,  OB,  OA',  OB'.  Then,  evidently,  because  OA'  -  OA,  0B'=  OB, 
R.ndAB'  =  AB,  the  triangles  OAB'  and  OAB  are  equal  and  similar. 
Hence  O  is  similarly  situated  with  regard  to  AB'  and  AB,  and 
is  therefore  one  and  the  same  point  of  the  plane  figure  in  its 
two  positions.  If,  for  the  sake  of  illustration, 
■we  actually  trace  the  angle  OAB  upon  the 
plane,  it  becomes  OAB'  in  the  second  posi- 
tion of  the  figure. 

92.   If  from  the  equal  angles  A' OB',  A  OB 

of  these  similar  triangles  we  take  the  corn- 

mon  part   A'OB,    we   have   the   remaining 

angles  AOA',  BOB'  equal,  and  each  of  them 

is  clearly  equal  to  the  angle  through  which 

"^  the  figure  must  have  turned  round  the  point  O 

to  bring  it  from  the  first  to  the  second  position. 

The  preceding  simple  construction  therefore  enables  us  not  only 

to  demonstrate  the  general  proposition  (§  91),  but  also  to  determine 

from  the  two  positions  of  one  line  AB,  A'B'  of  the  figure  the 

common  centre  and  the  amount  of  the  angle  of  rotation. 

93.  The  lines  equidistant  from  A  and  A',  and  from  B.  and  B\ 
are  parallel  \i AB  is  parallel  to  A'B'-,  and' therefore  the  construction 


KINEMATICS.  31 

fails,  the  point  O  being  infinitely 
distant,  and  the  theorem  becomes 
nugatory.  In  this  aise  the  motion  is 
in  fact  a  simple  translation  of  the 
figure  in  its  own  plane  without  rota- 
tion— since  as  AB  is  parallel  and  equal 
to  A'B',  -we  have  A  A'  parallel  and  equal  to  BB'  -,  and  instead  of 
there  being  one  point  of  the  figure  common  to  both  positions,  the 
lines  joining  the  successive  positions  of  every  point  in  the  figure  are 
equal  and  parallel. 

94.  It  is  not  necessary  to  suppose  the  figure  to  be  a  mere  tia& 
disc  or  plane — for  the  preceding  statements  apply  to  any  one  of  a 
set  of  parallel  planes  in  a  rigid  body,  moving  in  any  way  subject  to 
the  condition  that  the  points  of  any  one  plane  in  it  remain  always  ia 
a  fixed  plane  in  space. 

95.  There  is  yet  a  case  in  which  the  construction  in  §  Qf  ia 
nugatory — that  is  when  AA'  is  parallel  Tj 
X.o^BB',  but  AB  intersects  A'B'.  In 
this  case,  however,  it  is  easy  to  see  at 
once  that  this  point  of  intersection  is  the 
point  O  required,  although  the  former 
method  would  not  have  enabled  us  to 
find  it, 

96.  Very  many  interesting  applications  of  this  principle  may  be 
made,  of  which,  however,  few  belong  strictly  to  our  subject,  and  we 
shall  therefore  give  only  an  example  or  two.  Thus  v/e  know  that 
if  a  line  of  given  length  AB  move  with  its  extremities  always  in  two 
fixed  lines  OA,  OB,  any  point  in  it  as  P  describes  an  ellipse.  (This 
is  proved  in  §101  below.)  It  is  required  to  find  the  direction  of 
motion  of  P  at  any  instant,  i.  e.  to^  draw  a  tangent  to  the  ellipse. 
BA  will  pass  to  its  next  position  by  rotating  about  the  point  Q ;  found 
by  the  method  of  §  91  by  drawing  per- 
pendiculars to  OA  and  OB  at  A  and 
B.  Hence  B  for  the  instant  revolves 
about  Q,  and  thus  its  direction  of 
motion,  or  the  tangent  to  the  ellipse, 
is  perpendicular  to  QP.  Also  AB  in 
its  motion  always  touches  a  curve 
(called  in  geometry  its  envelop) ;  and 
the  same  principle  enables  us  to  find 

the  point  of  the  envelop  which  lies  in 

AlB,  for  the  motion  of  that  point  must    0  A 

evidently  be  ultimately  (that  is  for  a  very  smal'  displacement)  alon,:^ 
/IB,  and  the  only  point  which  so  moves  is  the  intersection  of  AB, 
.with  the  perpendicular  to  it  from  Q.  Thus  our  construction  would 
enable  us  to  trace  the  envelop  by  points. 


33  PRELIMINARY. 

97.  Again,  suppose  ABDCXo  be  a  jointed  frame,  AB  having' 
a  reciprocating  motion  about  A,  and  by  a  link  £D  turning  CD  ia 

the  same  plane  about  C.  Deter- 
mine the  relation  between  the 
angular  velocities  of  AB  and 
CZ)  in  any  position.  Evidently 
the  instantaneous  direction  of 
motion  of  B  is  transverse  to 
AB,  and  of  Z?  transverse  to  CD — 
hence  if  AB,  CD  produced  meet 

in  0,  the  motion  of  BD  is  for  an  instant  as  if  it  turned  about  O. 

From  this  it  may  easily  be  "seen  that  if  the  angular  velocity  of  AB 

be  w,   that  of  CD  is  -^^r^  -z=r^  w.    A  similar  process  is  of  course 
UJj  CD 

applicable  to  any  combination  of  machinery,  and  we  shall  find  it' very 

convenient  when  we  come  to  apply  the  principle  of  work  in  various 

problems  of  Mechanics. 

Thus  in  any  Lever,  turning  in  the  plane  of  its  arms — the  rate  ot 

motion  of  any  point  is  proportional  to  its  distance  from  the  fulcrum, 

and  its  direction  of  motion  at  any  instant  perpendicular  to  the  line 

joining  it  with  the  fulcrum.     This  is  of  course  true  of  the  particular 

form  of  lever  called  the  Wheel  and  Axle. 

98.  Since,  in  general,  any  movement  of  a  plane  figure  in  its  plane 
anay  be  considered  as  a  rotation  about  one  point,  it  is  evident  that 
two  such  rotations  may,  in  general,  be  compounded  into  oncr^  and 
therefore,  of  course,  the  same  may  be  done  with  any  number  of 
Totations.  Thus  let  A  and  B  be  the  points  of  the  i'gure  about  which 
in  succession  the  rotations  are  to  take  place.  By  rotation  about 
A,  B  is  brought  say  to  B",  and  by  a  rotation  about  B\.A  is  brought 
to  A'.  The  construction  of  §  91  gives  us  at  once  the  point  O  and 
the  amount  of  rotation  about  it  which  singly  gives  the  same  effect 
.as  those  about  A  and  B  in  succession.  But  there  is  one  case  of 
•e.xce])tion,  viz.  when  the  rotations   about  A   and  B  are  of  equal 

,_g  amount  and  in  opposite  directions.  In 
this  case  A'B'  is  evidently  parallel  and 
equal  to  AB,  and  therefore  the  com- 
pound result  is  a  translation  only.  Thus 
we  see  that  if  a  body  revolve  in  Succes- 
sion through  equal  angles,  but  in  oppo- 
site directions,  about  two  parallel  axes,  it  finally  takes  a  position 
to  which  it  could  have  been  brought  by  a  simple  translation  per- 
pendicular  to  the  lines  of  the  body  in  its  initial  or  final  position, 
which  were  successively  made  axes  of  rotation ;  and  inclined  to  their 
plane  at  an  angle  equal  to  half  the  supplement  of  the  common  angle 
•of  rotation. 

99.  Hence  to  compouitd  into  an  equivalent  rotation  a  rotation 
nnd  a  translation,  the  latter  being  eftected  parallel  to  the  plane  of 
the  former,  we  may  decompose  the  translation  into  two  rotations 


KINEMATICS. 


33 


Evidently  there  is  a  point 
the  space  through  which  the 


of  equal  amoutits  and  opposite  directions,  compound  one  of  them 

with  the  given  rotation  by  §  9S,  and  then  compound  the  other  with 

the  resuhant  rotation  by  the  same  process.     Or  we  may  adopt  the 

following  far  simpler  method  : — Let 

OA  be  the  translation  common  to 

all    points   in   the    plane,    and    let 

BOC   be    the  .angle    of    rotation 

about  (9,  BO  being  drawn  so  that 

OA  bisects  the  exterior  angle  COB'. 

B'  in  BO  produced,  such  that  B C, 

rotation  carries  it,  is  equal  and  opposite  to  OA.     This  point  retains 

its  former  position  after  the  perform.ance  of  the  compound  operation ; 

so*that  a  rotation  and  a  translation  in  one  plane  can  be  compounded 

into  an  equal  rotation  about  a  difterent  axis. 

100.  Any  motion  whatever  of  a  plane  figure  in  its  own  plane 
might  be  produced  by  the  rolling  of  a  curve  fixed  to  the  figure 
upon  a  curve  fixed  in  the  plane. 

For  we  may  consider  the  whole  motion  as  made  up  of  successive 
elementary  displacements,  each  of  which 'corresponds,  as  we  have 
seen,  to  an  elementary  rotation  about  some 
point  in  the  plane.  Let  6*,,  O^,  O^,  etc., 
be  the  successive  points  of  the  figure  about 
•which  the  rotations  take  place, -(^j,  o„,  0.^, 
etc.,  the  positions  of  these  points  on  the 
plane  when  each  is  the  instantaneous  centre 
of  rotation.  Then  the  figure  rotates  about 
(?,  (or  <7j,  which  coincides  with  it)  till  O^ 
coincides  with  <?„,  then  about  the  latter  till 
O^  coincides  with  0^,  and  so  on.     Hence,  if 

v/e  join  Cj,  Cj,  O^,  etc.,  in  the  plane  of  the  figure,  and  0^ ,  (?,,  0^^  etc., 
in  the  fixed  plane,  the  motion  will  be  the  same  as  if  the  polygon 
0^0  ^O^y  etc,  rolled  upon  the  fixed  polygon  o^o.^o.,^,  etc. '  By 
supposing  the  successive  displaceivients  small  enough,  the  sides 
of  the-se  polygons  gradually  diminish,  and  the  polygons  finally 
become  continuous  curves.     ILnce  the  theorem. 

From  this  it  immediately  follows,  that  any  displacement  of  a  rigid 
solid,  whicli  is  in  directions  wholly  perpendicular  to  a  fixed  line, 
may  be  produced  by  the  rolling  of  a  cylinder  fixed  in  tlie  solid  on 
another  cylinder  fixed  in  space,  the  axes  of  the  cylinders  being 
parallel  to  the  fixed  line. 

101.  As  an  interesting  example  of  this  theoreni,  let  us  recur  to  the 
tase  of  §  96  : — A  circle  may  evidently  be  circumscribed  about  OBQA; 
and  it  must  be  of  invariable  magnitude,  since  in  it  a  chord  of  given 
length  z:?^  subtends  a  given  angle  (9  at  the  circumference.  Also  OQ  is 
a  diameter  of  this  circle,'  and  is  therefore  constant.  Hence,  as  Q  is 
momentarily  at  rest,  the  motion  of  the  circle  circumscribing  OBi^A 
is  one  of  internal  rolling  on  a  circle  of  double  its  diameter.  Hence 
if  a  circle  roll  internally  on  another  of  twice  its  diameter  any  point  in 


34 


PRELlMmARY. 


its  circumference  describes  a  diameter  of  the  ifixed  circle,  any  ctRer 
point  in  its  plane  an  ellipse.  This  is  precisely  the  same  proposition 
as  that  of  §  86,  although  the  ways  of  arriving  at  it  are  very  different. 

102.  We  may  easily  employ  this  result,  to  give,  the  proof,  promised 
in  \  96,  that  the  point  P  oi  AB  describes  an  ellipse.  Thus  let 
OA,  OB  be  the  fixed  lines,  in  which  the  extremities,  of  AB  move. 
Draw  the  circle  AOBD,  circumscribing  ^4  (7^,  and.  let  CD  be  the 
diameter  of  this  circle  which  passes  through  P.  \  While  the  two 
points  A  and  B  of  this  circle  move  alon^  OA  and  OB,  the  points 
C  and  D  must,  because  of  the  invariability  of  the  angles  BOD, 

AOC,  move  along  straightiines  OC, 
CD,  and  these  are  evidently  at  right 
angles.  Hence  the  path  of  P  may 
be  considered  as  that  of  a  point  in 
a  line  whose  ends  move  on  two 
mtitually  perpendicular  lines.  Let  £ 
be  the  centre  of  the  circle;  join  OE, 
and  produce  it  to  meet,  in  F,  the 
line  FFG  drawn  through  P  parallel 
to  DO.  Then  evidently  EF=EP, 
hence  F  describes  a  circle  about  O. 
Also  FP  \FG  ::  2FE  :  FO,  or  PG 
is  a  conb-?.nt  submultiple  oi  EG;  and 
therefore  the  locus  of  P  is  an  ellipse, 
whose  major  axis  is  a  diameter  of  the 
circular  path  of/!  Its  semi-axes  are  Z?/' along  OC,  and  PC  along  OD. 

103.  When  a  circle  rolls  upon  a  straight  line,  a  point  in  its 
circumference  describes  a  Cycloid,  an  internal  point  describes  a 
Prolate  Cycloid,  an  external  point  a  Curtate  -  Cycloid.  The  two 
latter  varieties  are  sometimes  called  Trochoids. 

The  general  form  of  these  curves  will  be  seen  in  the  succeeding 
figures ;  and  in  what  follows  we  shall  confine  our  remarks  to  the 
cycloid  itself,  as  it  is  of  greater  consequence  than  the  others.  The 
next  section  contains  a  simple  investigation  of  those  properties  of 
the  cycloid  which  are  most  useful  in  our  subject. 


KINEMATICS, 


35 


104.  Let  AB  be  a  diameter  of  the  generating  (or  rolling)  circle, 
EC  the  line  on  which  it  rolls.  The  points  A  and  B  describe  similar 
and  equal  cycloids,  of  which  AQ_C  and  BS  are  portions.  \i PQR 
be  any  subsequent  position  of  the  generating  circle,  Q  and  S  the 
new  positions  of  A  and  B,  QFS  is  of  course  a  right  angle.  If, 
therefore,  QR  be  drawn  parallel  to 
PS,  PR  is  a  diameter  of  the  rolling 
circle,  and  R  lies  in  a  straight  line 
AH  drawn  parallel  to  BC.  Thus 
AR  =  BP.  Produce  QR  to  T, 
rmklng  RT=  QR  =  PS.  Evidently 
the  curve  AT,  whicli  is  the  locus 
cf  T,  is  similar  and  equal  to  BS, 
and  is  therefore  a  cycloid  similar  and 
equal  to  AC.  But  QR  is  perpen- 
dicular to  PQ,  and  is  therefore  the 
instantaneous  direction'  of  motion  of 
Q,  or  is  the  tangent  to  the  cycloid 

AQC.     Similarly,  PS  is  perpendicular  to  the  cycloid  BS  at  ^9. 
therefore  TQ  is  perpendicular  to  y/ J"  at  T.     ~^ 
the  evolute  of  AT,  and  arcAQ=  QT=2QR, 

105.  When  a  circle  rolls  upon  an- 
other circle,  the  curve  described  by  a 
point  in  its  circumference  is  called  an 
Epicycloid,  or  a  Hypocycloid,  as  the 
rolling  circle  is  without  or  within  the 
ftxed  circle;  and  when  the  tracing-point 
is  not  in  the  circumference,  we  have 
Epitrochoids  and  H}'potTOchoids.  Of 
^he  latter  classes  we  have  already  met 
tt-ith  examples  (§§  87,  loi),  and  others 
will  be  presently  mentioned.  Of  the 
former  we  have,  in  the  first  of  the 
appended  figures,  the  case  of  a  circle 
rolling  externally  on  another  of  equal 
size.  The  curve  in  this  case  is  called 
the  Cardioid. 

In  the. second  figure  a  circle  rolls  ex- 


and 
Hence  (§  22)  AQC  is 


36 


PRELIMINARY, 


ternally  on  another  of  twice  its  radius.    The  epicyclol'd  so  jdescribed 

is  of  importance  in  optics,  and  will,  with  others,  be  referred  to  when 

we  consider  the  subject  of  Caustics  by  reflexion. 

In  the  third  figure  we  have  a  hypo- 
cycloid  traced  by  the  rolling  of  one 
circle  internally  on  another  of  four 
times  its  radius. 

The  curve  of  §  87  is  a  hypotrochoid 
described  by  a  point  in  the  plane  of  a 
circle  which  rolls  internally  on  another 
of  rather  more  than  twice  its  diameter, 
the  tracing-point  passing  through  the 
centre  of  the  fixed  circle.  Had  the 
diameters  of  the  circles  been  exactly  a? 
I  :  2,  §  loi  shows  us  that  this  curve 

would  have  been  reduced  to  a  single  straight  line. 

106.  If  a  rigid  body  move  m  any  way  whatever,  subject  only  to 
the  condition  that  one  of  its  points  remains  fixed,  there  is  always 
(without  exception)  one  line  of  it  through  this  point  common  to  the 
body  in  any  two  positions. 

Consider  a  spherical  surface  within  the  body,  v/ith  its  centre  at  the 
fixed  point  C.  All  points  of  this  sphere  attached  to  the  body  will 
move  on  a  sphere  fixed  in  space.  Hence  the  construction  of  §  91 
may  be  made,  only  with  great  circles  instead  of  straight  Sines ; 
and  the  same  reasoning  will  apply  to  prove  that  the  point  O  t^ius 
obtained  is  common  to  the  body  in  its  two  positions.  Hence  every 
point  of  the  body  in  the  line  OC,  joining  0  with  the  fixed  point, 
must  be  common  to  it  in  the  tNvo  positions.  Hence  the  body  may 
pass  from  any  one  position  to  any  other  by  a  definite  amount  of 
rotation  about  a  definite  axis.  And  hence,  also,  successive  or  sirAul- 
taneous  rotations  about  any  number  of  axes  through  the  fixed  point 
may  be  compounded  into  one  such  rotation. 

107.  Xet  OA,  OB  be  two  axes  about  which  a  body  revolves  with 
angular  velocities  t!>,  w^  respectively. 

With  radius  unity  describe  the  arc  AB,  and  in  it  take  any  point  /. 
Draw  /a,  //3  perpendicular  to  OA,  OB  respectively.  Let  tlie  rota- 
tions about  the  two  axes  be  such  that  that  about 
OB  tends  to  raise  I  above  the  plane  of  the 
paper,  and  that  about  OA  to  depress  it.  In  an 
infinitely  short  interval  of  time  t,  the  amounts  of 
these  displacements  will  be  w^/yg .  t  and  —  w/a .  t.' 
The  point  /,  and  therefore  every  point  in  the 
line  01,  will  be  at  rest  during  the  interval  t  if 
the  sum  of  these  displacements  is  zero — i.e.  if 
Wj .  //3  =  w   /a. 

Hence  the  line  01  is  mstantaneously  at  rest,  or  the  two  rotations 
about  OA  and  OB  may  be  compounded  into  one  about  01.     Draw 


kinematics:  37- 

tp,  Tq,  parallel  to  OB,  OA  respectively.    Then,  ejcpressing  in  two 
ways  the  area  of  the  parallelogram  IpOq,  we  have 

Hence  Oq  -.    Op  ::  'i>,   :  w. 

In  words,  if  on  the  axes  OA,  OB,  we  measure  off  from, 6*  lines 
Op,  Oq,  proportional  respectively  to  the  angular  velocities  about 
these  axes — the  diagonal  of  the  parallelo^:jrara  of  which  tliese  are 
contiguous  sides  is  the  resultant  axis. 

Again,  if  Bb  be  drawn  perpendicular  to  OA,  and  if  H  be  tbe 
angular  velocity  about  01,  the  whole  displacement  of  i?  may  evidently 
be  represented  either  by 

^.Bb  oxO..  13. 
Hence  C!  :  oj  ::  B'>  :  7/3 

:•   01  :   Op. 

And  thus  on  the  scale  on  which  Op,  Oq  represent  the  component 
angular  velocities,  the  diagonal  01  represents  their  resultant. 

108.  Hence  rotations  are  to  be  compounled  according  to  t>.e 
same  law  as  velocities,  and  therefore  'the  single  angular  velocity, 
equivalent  to  three  co-existent  angular  velocities  about  three  mutually 
perpendicular  axes,  is  determined  in  magnitude,  and  the  du-ection 
of  its  axis  is  found,  as  follows : — The  square  of  the  resultant  angular 
velocity  is  the  sum  of  the  squares  of  its  components,  and  the  ratios 
of  the  three  components  to  the  resultant  are  the  direction-cosines 
of  the  axis. 

Hence  also,  an  angular  velocity  about  any  lin2  may  be  resolved 
into  three  about  any  set  of  rectangular  lines,  the  resolution  in  each 
case  being  (like  that  of  simple  velocities)  effected  by  multiplying  by 
the  cosine  of  the  angle  between  the  directions. 

Hence,  just  as  in  §  38  a  uniform  acceleration,  acting  perpendi- 
cularly to  tlie  direction  of  motion  of  a  point,  produces  a  change  in 
the  direction  of  motion,  but  does  not  influence  the  vdccily ;  so.  if 
a  body  be  rotating  about  an  avis,  and  be  sulijected  to  an  action 
tending  to  produce  rotation  about  a  perpendicular  axis,  the  result 
will  be  a  change  of  direction  of  the  axis  about  which  the  body 
revolves,  but  no  change  in  the  angular  velocity.  On  this  kinematical 
principle  is  founded  the  dynamical  explanation  of  th.e  precession  of 
the  equinoxes,  and  some  of  the  seemingly  marvellous  performances 
of  gyroscopes  and  gyrostats. 

109.  li  a  pyramid  or  cone  of  any  form  roll  on  a  similar  pyramid 
(the  image  in  a  plane  mirror  of  the  first  position  of  the  first)  all 
round,  it  clearly  comes  back  to  its  primitive  position.  This  (as  all 
rolling  of  cones)  is  exhibited  best  by  taking  the  intersection  of  each 
with  a  spherical  surface.  Thus  we  see  that  if  a  s};herical  pol\g; on 
turns  about  its  angular  points  in  succession,  ahva\s  keej'ing  on  t:;e 
Spherical  surface,  and  if  the  angle  through  which  it  turns  a'i;'j;u  tich 
point  is  twice  the  supplement  of  the  angle  of  the  poh-gon,  or,  '^--hich 


3$  PRELIMINARY. 

fwill  come  to  the  same  thing,  if  it  be  in  the  other  direction,  but. 
equal  to  twice  the  angle  itself  of  the  polygon^  it  will  be  brought  to 
its  original  position. 

110.  The  method  of  §  loo  also  applies  to  the  case  of  §  io6;  and 
it  is  thus  easy  to  show  that  the  most  general  motion  of  a  spherical 
figure  on  a  fixed  spherical  surface  is  obtained  by  the  rolling  of  a 
cur.ve  fixed  in  the  figure  on  a  curve  fixed  on  the  sphere.  Hence  as 
at  each  instant  the  line  joining  C  and  O  contains  a  set  of  points  of 
the  body  v.-hich  are  momentarily  at  rest,  the  most  general  motion  of 
a  rigid  body  of  which  one  point  is  fixed  consists  in  the  rolling  of  a 
cone  fixed  in  the  body  upon  a  cone  fixed  in  space — the  vertices  of 
both  being  at  the  fixed  point. 

111.  To  complete  our  kinematical  investigation  of  the  motion  of 
a  body  of  which  one  point  is  fixed,  we  require  a  solution  of  the  fol- 
lowing problem: — From  the  given  angular  velocities  of  the  body 
at  each  instant  about  three  rectangular  axes  attached  to  it  to  de- 
termine the  position  of  the  body  in  space  after  a  given  time.  But 
the  general  solution  of  this  problem  demands  higher  analysis  than 
can  be  admitted  into  l^e  present  treatise. 

112.  We  shall  next  consider  the  most  general  possible  motion  of 
a  rigid  body  of  which  no  point  is  fixed — and  first  we  must  prove 
the  following  theorem.  There  is  one  set  of  parallel  planes  in  a 
rigid  body  which  are  parallel  to  each  other  in  any  two  positions  of 
the  body.  The  parallel  lines  of  the  body  perpendicular  to  these 
planes  are  of  course  parallel  to  each  other  in  the  two  positions. 

Let  C  and  C  be  any  point  of  the  body  in  its  first  and  second 
positions.  Move  the  body  without  rotation  from  its  second  position 
to  a  third  in  which  the  point  at  C  in  the  second  position  shall 
occupy  its  original  posidon  C.  The  preceding  demonstration  shows 
that  there  is  a  line  CO  common  to  the  body  in  its  first  and  third 
positions.  Hence  a  line  CO'  of  the  body  in  its  second  position  is 
parallel  to  the  same  line  CO  in  the  first  position.  This  of  course 
clearly  applies  to  every  line  of  the  body  parallel  to  CO,  and  the 
planes  perpendicular  to  these  lines  also  remain  parallel. 

113.  Let  ^►S'  denote  a  plane  of  the  body,  the  two  positions  of  which 
are  parallel.  Move  the  body  from  its  first  position,  without  rotation, 
in  a  direction  perpendicular  to  6",  till  S  comes  into  the  plane  of  its 
second  position.  Then  to  get  the  body  into  its  actual  position,  such 
a  motion  as  is  treated  in  §  91  is  farther  required.  But  by  §  91  this 
may  be  effected  by  rotation  about  a  certain  axis  perpendicular  to  the 
plane  S,  unless  the  motion  required  belongs  to  the  exceptional  case 
of  pure  translation.  Hence  (this  case  excepted),  the  body  may 
be  brought  from  the  first  position  to  the  second  by  translation 
through  a  determinate  distance  perpendicular  to  a  given  plane,  and 
rotation  through  a  determinate  angle  about  a  determinate  axis  per-' 
pendicular  to  that  plang.  This  is  precisely  the  motion  of  a  screw 
in  its  nut. 


KINEMATICS.  39 

114.  To  understand  the  nature  of  this  motion  we  may  com- 
mence ivith  the  shding  of  one  straight-edged  board  on  another. 

Thus  let  CDEF  be  a  plane  board  whose  edge,  DE,  slides  on 
the  edge,  AB,  of  another  board,  ABQ,  of  which  for  convenience 
we  suppose  the  edge,  AC,  to  be  hori- 
zontal. By  §  30,  if  the  upper  board 
move  horizontally  to  the  right,  the 
constraint  will  give  it,  in  addition,  a 
vertically  upward  motion,  and  the  rates 
of  these  motions  are  in  the  constant 
ratio  of  /I  C  to  CB.  Now,  if  both 
planes  be  bent  so  as  to  form  portions 
of  the  surface  of  a  vertical  right  cylinder,  the  motion  of  DF  parallel 
to  ^C  will  become  a  rotation  about  the  axis  of  the  cylinder,  and 
the  necessary  accompaniment  of  vertical  motion  \vill  remain  un- 
changed. As  it  is  evident  that  all  portions  of  AB  will  be  equally 
inclined  to  the  axis  of  the  cylinder,  it  is  ob\ious  that  the  thread 
of  the  screw,  which  corresponds  to  the  edge,  DE,  of  the  upper 
board,  must  be  traced  on  the  cylinder  so  as  always  to  make  a  con- 
stant angle  with  its  generating  lines  (^  128).  A  hollov/  mould 
taken  from  the  screw  itself  forms  v/hat  is  called  the  nut — the  re- 
presentative of  the  board,  ABC — and  it  is  obvious  that  the  screw 
cannot  move  without  rotating  about  its  axis,  'if  the  nut  be  fixed. 
If  a  be  the  radius  of  the  cylinder,  w  the  angular  velocity,  a  the 
inclination  of  the  screw  thread  to  a  generating  line,  u  the  linear 
velocity  of  the  axis  of  the  screw,  we  see  at  once  from  the  above  con- 
struction that 

a^  \  ti  w  AC  :  CB  ::  sin  a  :  cOS  a, 

which  gives  the  requisite  relation  between  w  and  tt. 

115.  In  the  excepted  case  of  §  113,  the  whole  motion  consists 
of  two  translations,  which  can  of  course  be  compounded  into  a 
single  one  :  and  thus,  in  this  case,  there  is  no  rotation  at  all,  or 
every  plane  of  it  fulfils  the  specified  condition  for  6""  of  §  113. 

116.  We  m.ay  now  briefly  consider  the  case  in  whicli  the  guiding 
cones  (§  no)  are  both  circular,  as  it  has  important  applications  to 
the  motion  of  the  earth,  the  evolutions  of  long  or  flattened  projec- 
tiles, the  spinning  of  tops  and  g-yroscopes,  etc.  The  motion  in  diis 
case  may  be  called  Precessional  Rotation.  The  plane  through  the 
instantaneous  axis  and  the  axis  of  the  fixed  cone  passes  through  the 
axis  of  the  rolling  cone.  This  plane  turns  round  the  axis  of  the 
fixed  cone  with  an  angular  velocity  fi,  which  must  clearly  bear  a 
constant  rado  to  the  angular  velocity  w  of  the  rigid  body  about  its 
instantaneous  axis. 

117.  The  motion  of  the  plane  containing  these  axes  is  called  the 
precession  in  any  such  case.  What  we  have  denoted  by  fi  is  the 
angular  velocity  of  the  precession,  or,  as  it  is  sometimes  called,  the 
rate  of  precession. 


40  PRELIMINARY. 

The  angular  motions  «,  O  are  to  one  another  inversely  as  the 
distances  of  a  point  in  the  axis  of  the  rolling  cone  from  the  io- 
stantaneous  axis  and  from  the  axis  of  the  fixed  cone. 

For,  let  OA  be.the  axis  of  the  fijed  cone,  OB  that  of  the  rolling 
cone,  and  01  the  instantaneous  axis.  From  any  point  P  in  OB 
draw  PN  perpendicular  to  01,  and  PQ  perpendicular  to  OA. 
Then  we  perceive  that  P  moves  always  in  the  circle  whose  centre 
is  Q.,  radius  PQ,  and  plane  perpendicular 
to  OA.  Hence  the  actual  velocity  of 
7  the  point  /*  is  U .  QP.     But,  by  the  prin- 

}{/  ciples  explained  above  (§  no)  the  velocity 

y<  \         »     of  P  is  the  same  as  that  of  a  point  moving 

.,....../.... "■}p^        in  a  circle  whose  centre  is  N,  plane  per- 

/     ^^^^  "  pendicular  to  ON,  and  radius  NP,  which, 

/  .^  as  this  radius  revolves  with  angular  velo- 

"^  city  to,  is  <o .  NP.    Hence 

^.QP^i^.NP, 
or        to  :  fi  ::  QP  :  NP. 

118.  Suppose  a  rigid  body  bounded  by  any  curved  surface  to  be 
touched  at  any  point  by  another  such  body.  Any  motion  of  one 
on 'the  other  must  be  of  one  or  more  of  the  forms  sliding,  rolling, 
or  spinning.  The  consideration  of  the  first  is  so  simple  as  to  require 
no  comment. 

Any  motion  in  v/hich  the  bodies  have  no  relative  velocity  at  the 
point  of  contact,  must  be  rolling  or  spinning,  separately  or  combined. 

Let  one  of  the  bodies  rotate  about  successive  instantaneous  axes, 
all  lyii'ig  in  the  comnion  tangent  plane  at  the  point  of  instantaneous 
contact,  and  each  passing  through  this  point — the  other  body  being 
fixed.  This  motion  is  what  we  call  rolling,  or  simple  rolling,  of  the 
movable  body  on  the  fixed. 

On  the  other  hand,  let  the  instantaneous  axis  of  the  moving  body 
-be  the  com.mon  normal  at  the  point  of  contact.  This  is  pure  spin- 
:nirig,  and  does  not  change  the  point  of  contact. 

Let  it  move,  so  that  the  instantaneous  axis,  still  passing  through 
the  point  of  conlact,'  is  neither  in,  nor  perpendicular  to,  the  tangent 
plane.     This  ruction  is  combined  rolling  and  spinning. 

119.  As  an  example  of  pure  rolling,  we  may  take  that  of  one 
cylinder  on  another,  the  axes  being  parallel. 

Let  p  be  the  radius  of  curvature  of  the  rolling,  a  of  the  fixed, 
cylinder ;  <o  the  angular  velocity  of  the  former,  V  the  linear  velocity 
of  tl-ie  point  of  contact.    We  have 

For,  in"  the  figure,  suppose  P  to  be  at  any  time  the  point  of 
contact,  and  Q  and  p  the  points  which  are  to  be  in  contact  after 
a  very,  small  interval  t;  (9,  (7  the  centres  of  curvature  ;^ /'(?/>  =  ^, 
POQ^^. 


KINEMATICS.  41 

Then  PQ  -Pp-=  space  described  by  point  of  con- 
tact.    In  svmbols 

p4i=^aO=  Ft. 

Also,  before  (/Q  and  OP  can  coincide  in  direc- 
tion, the  former  must  evidently  turn  through  an  angle 

Therefore  cor  =  5  4-  9 ; 

and  by  eliminating  6  and  <^,  and  dividing  by  t,   we 
get  the  above  result. 

It  is  to  be  understood  here,  that  as  the  radii  of 
curvature,  have  been  considered  posiuve  when  both 
surfaces  are  convex,  the  negative  sign  must  be  intro- 
duced for  either  radius  when  the  corresponding  sur- 
face is  concave. 

Hence  the  angular  velocity  of  the  rolling  curve  is  in  this  case 
e'qual  to  the  product  of  the  linear  velocity  of  the  point  of  contact  into 
the  sum  or  difference  of  the  curvatures,  according  as  the  curves  are 
both  convex,  or  one  concave  and  the  other  convex. 

120.  We  may  now  take  up  a  few  points  connected  with  the  curva- 
ture of  surfaces,  which  are  useful  in  various  parts  of  our  .jubject. 

The  tangent  plane  at  any  point  of  a  surface  may  or  may  not  cut 
it  at  that  point.  In  the  former  case,  the  surface  bends  away 'from 
the  tangent  plane  partly  towards  one  side  of  it,  and  partly  towards 
the  other,  and  has  thus,  in  some  of  its  normal  sections,  curvatures 
oppositely  directed  to  those  in  others.  In  the  latter  case,  the  sur- 
face on  every  side  of  the  point  bends  away  from  the  same  side  of 
its  tangent  plane,  and  the  curvatures  of  all  normal  sections  are 
similarly  directed.  Thus  we  may  divide  curved  surfaces  into  Anfi- 
clastic  and  Synclastic.  A  saddle  gives  a  good  example  of  the 
foruier  class ;  a  ball  of  the  latter.  Curvatures  in  opposite  directions, 
with  reference  to  the  tangent  plane,  have  of  course  different  signs. 
The  outer  portion  of  the  surface  of  an  anchor-ring  is  synchstic,  the 
inner  anticlastic 

121.  Meunicr's  Theorem, — ^The  cur\-ature  of  an  oblique  section 
of  a  surface  is  equal  to  that  of  the  normal  section  through  the  same 
tangent  line  multiplied  by  the  secant  of  the  inclination  of  the  planes 
of  the  sections.  This  is  evident,  from  the  most  elementary  con- 
siderations regarding  projections. 

122.  Eulcr's  Theorem. — There  are  at  every  point  of  a  synchstic 
surface  two  normal  sections,  in  one  of  which  the  curvature  is  a 
maximum,  in  the  other  a  minimum  \  and  these  are  at  right  angles 
to  each  other. 

In  an  anticlastic  surface  there  is  maximum  curvature  (but  in 
opposite  directions)  in  the  two  normal  sections  whose  planes  bisect 
the  angles  between  the  Unes  in  which  the  surface  cuts  its  tangent 

A't.l.  2:5—3 


42  PRELIMINARY. 

plane.    On  account  of  the  difference  of  sign,  these  may  be  con- 
sidered as  a  maximum  and  a  minimum. 

Generally  the  sum  of  the  curvatures  at  a  point,  in  any  two  normal 
planes  at  right  angles  to  each  other,  is  independent  of  the  position 
of  these  planes." 

If  --  and  -  be  the  maximum  and  minimum  curvatures  at  any 

p  <7 

point,  the  curvature  of  a  normal  section  making  an  angle  $  with  the 
norn;al  section  of  maximum  curvature  is 

•  -  cos'  $  +  -  sin'  0, 
p  <r 

which  includes  the  above  statements  as  particular  cases. 

123.  Let  P,  p  be  two  points  of  a  surface  indefinitely  near  to  each 
other,  and  let  r  be  the  radius  of  curvature  of  a  normal  section  passing 
through  them.  Then  the  radius  of  curvature  of  an  obhque  section 
through  the  same  points,  incUned  to  the  former  at  an  angle  a, 
is  rcos  a  (§  i2i).  /Jso  the  length  along  the  normal  section,  from 
P  to  /,  is  less  than  that  along  the  oblique  section — since  a  givert" 
chord  cuts  oflf  an  arc  from  a  circle,  longer  the  less  is  the  radius 
of  that  circle. 

124.  Hence,  if  the  shortest  possible  line  be  drawn  from  one  point 
of  a  surface  to  another,  its  osculating  plane,  or  plane  of  curvature, 
is  everywhere  perpendicular  to  the  surface. 

Such  a  curve  is  called  a  Geodetic  line.  And  it  is  easy  to  see  that 
it  is  the  line  in  which  a  flexible  and  inextensible  string  would  touch 
the  surface  if  stretched  between  those  points,  the  surface  being  sup- 
posed smooth. 

125.  A  perfectly'  flexible  but  inextensible  surface  is  suggested, 
although  not  realized,  by  paper,  thin  sheet-metal,  or  cloth,  when  the 
surface  is  plane ;  and  by  sheaths  of  pods,  seed-vessels,  or  the  like, 
when  not  capable  of  being  stretched  flat  without  tearing.  The  process 
of  changing  the  form  of  a  surface  by  bending  is  called  *  developing? 
But  the  term.  ^ Detielopable  Surface^  is  commonly  restricted  to  such 
inextensible  surfaces  as  can  be  developed  into  a  plane,  or,  in  com- 
mon language,  'smoothed  flat.' 

126.  The  geometry  or  kinematics  of  this  subject  is  a  great  contrast 
to  that  of  the  flexible  line  (§  i6),  and,  in  its  merest  elements,  presents 
ideas  not  very  easily  apprehended,  and  subjects  of  investigation  that 
have  exercised,  and  perhaps  even  overtasked,  .the  powers  of  some 
of  the  greatest  mathematicians. 

127.  Some  care  is  required  to  form  a  correct  conception  of  what 
is  a  perfectly  flexible  inextensible  surface.  First  let  us  consider  a 
plane  sheet  of  paper.  It  is  very  flexible,  and  we  can  easily  form 
Ihe  conception  from  it  of  a  sheet  of  ideal  matter  perfectly  flexible. 


KINEMATICS.  43 

It  5S  very  inextensible ;  that  is  to  say,  it  yields  very  little  to  any 
application  of  force  tending  to  pull  or  stretch  it  in  any  direction, 
up  to  the  strongest  it  can  bear  without  tearing.  It  does,  of  course, 
stretch  a  little.  It  is  easy  to  test  that  it  stretches  when  under  the 
influence  of  force,  and  that  it  contracts  again  when  the  force  is 
removed,  although  not  always  to  its  original  dimensions,  as  it 
may  and  generally  does  remain  to  some  sensible  extent  permanently 
stretched.  Also,  flexure  stretches  one  side  and  condenses  the  other 
temporarily;  and,  to  a  less  extent,  permanently.  Under  elasticity  we 
may  return  to  this.  In  the  meantime,  in  considering  illustrations  of 
our  kinematical  propositions,  it  is  tiecessary  to  anticipate  such  phy- 
sical circumstances. 

128.  The  flexure  of  an  inextensible  surface  which  can  be  plane, 
is  a  subject  which  has  been  well  worked  by  geometrical  investigators 
and  writers,  and,  in  its  elements  at  least,  presents  little  difficulty.  The 
first  elementary  conception  to  be  formed  is,  that  such  a  surface  (if 
perfectly  flexible),  taken  plane  in  the  first. place,  may  be  bent  about 
any  straight  line  ruled  on  it,  so  that  the  two  plane  parts  may  make 
any  angle  v.-ith  one  another. 

Such  a  line  is  called  a  'generating  line'  of  the  surface  to  be 
formed. 

Next,  we  may  bend  one  of  these  plane  parts  about  any  other  line 
which  does  not  (within  the  limits  of  the  sheet)  intersect  the  former ; 
and  so  on.  If  these  lines  are  infinite  in  number,  and  the  angles  of 
bending  infinitely  small,  but  such  that  their  sum  may  be  finite,  we 
have  our  plane  surface  bent  into  a  curved  surface,  which  is  of  course 
'developable'  (§  125). 

129.  Lift  a  square  of  paper,  free  from  folds,  creases,  or  ragged 
edges,  gently  by  one  corner,  or  otherwise,  without  crushing  or  forcing 
it,  or  very  gently  by  two  points.  It  will  hang  in  a  form  which  is  very 
rigorously  a  developable  surface ;  for  although  it  is  not  absolutely 
inextensible,  yet  the  forces  which  tend  to  stretch  or  tear  it,  when  it 
is  treated  as  above  described,  are  small  enough  to  produce  absolutely 
no  sensible  stretching.  Indeed  the  greatest  stretching  it  can  expe- 
rience without  tearing,  in  any  direction,  is  not  such  as  can  affect  the 
form  of  the  surface  much  when  sharp  llexures,  singular  points,  etc., 
are  kept  clear  ofiF. 

130.  Prisms  and  cylinders  (v/hen  the  lines  of  bending,  §  128, 
are  parallel,  and  finite  in  number  with  finite  angles,  or  infinite 
in  number  with  infinitely  sm.all  angles),  and  pyramids  and  cones 
(the  lines  of  bending  meeting  in  a  point  if  produced),  are  clearly 
included. 

131.  If  the  generating  lines,  or  line-edges  of  the  angles  of  bending, 
are  not  parallel,  they  must  meet,  since  they  are  in  a  plane  when  the 
surface  is  plane.  If  they  do  not  meet  all  in  one  point,  they  must* 
meet  in  several  points :  in  general,  each  one  meets  its  predecessor 
and  its  successor  in  different  points. 


44 


PRELIMINARY. 


There  is  still  no  difficulty  in  understanding  the  form  of,  say  a 
or  circle,  of  the  plane  surface  when  bent  as  explained  above, 
provided  it  does  not  include  any  of  these  points 
of  intersection.  When  the  number  is  infinite, 
and  the  surface  finitely  curved,  the  developable 
lines  will,  in  general,  be  tangents  to  a  curve  (the 
locus  of  the  points  of  intersection  when  the 
number  is  innnite).  '  This  curve  is  called  the 
edge  of  regression.  The  surface  must  clearly, 
when  complete  (according  to  mathematical  ideas), 
consist  of  two  sheets  meeting  in  this  edge  of 
regression  (just  as  a  cone  consists  of  two 
sheets  meeting  in  the  vertex),  because  each 
tangent  may  be  produced  beyond  the  point 
of  contact,  instead  of  stopping  at  it,  as  in  the  preceding  diagram. 

133.  To  construct  a  complete  developable  surface  in  two  sheets 
from  its  edge  of  regression — 

Lay  one  piece  of  perfectly  flat,  un- 
wrinkled,  smooth-cut  paper  on  the  top 
of  another.     Trace   any  curve   on.  the 
other,   and  let  it  have  no  point  of  va 
flection,  but  everywhere  finite  curvature 
Cut  the  paper  quite  away  on,  the  con- 
cave side.     If  the  curve  traced  is  closed 
it  must  be  cut  open  (see  second  diagram) 
The  limits  to   the  extent  that  may  be  left  uncut  away,  are  the 
tangents  drawn  outwards  from  the  two  ends,  so  that,  in  short,  no 
portion  of  the  paper  through  which  a  real  tangent  does  not  pass 
is  to  be  left. 

Attach  the  two  sheets  together  by  very  slight  paper  qr  muslin 
clamps  gummed  to  them  along  the  common  curved  edge.  These 
must  be  so  slight  as  not  to  interfere  sensibly  with 
the  flexure  of  the  two  sheets.  Take  hold  of  one 
corner  of  one  sheet  and  lift  the  whole.  The  two 
will  open  out  into  two  sheets  of  a  developable 
surface,  of  which  the  curve,  bending  into  a  tor- 
tuous curve,  is  the  edge  of  regression.  The  tan- 
gent to  the  curve  drawn  in  one  direction  from 
the  point  of  contact,  will  always  lie  in, one  of  the 
sheets,  and  its  continuation  on  the  other  side  in  the 
other  sheet.  Of  course  a  double-sheeted  developable  polyhedron  can 
be  constructed  by  this~process,  by  starting  from  a  polygon  instead 
of  a  curve. 

134.  A  flexible  but  perfectly  inextensible  surface,  altered  in  form 
in  any  way  possible  for  it,  must  keep  .any  line  traced  on  it  un- 
changed in  length ;  and  hence  any  two  intersecting  lines  unchanged 
in  mutual  inclinatioa  Hence,  also,  geodetic  lines  must  remain 
geodetic  lines. 


KINEMATICS.  45 

135.  We  have  now  to  consider  the  very  important  kinematical 
conditions  presented  by  the  changes  of  volume  or  figure  experienced 
by  a  solid  or  liquid  mass,  or  by  a  group  of  points  whose  positions 
with  regard  to  each  other  are  subject  to  known  conditions. 

Any  such  definite  alteration  oi  form  or  dimensions  is  called  a 
Strain. 

Thus  a  rod  which  becomes  longer  or  shorter  is  strained.  Water, 
when  compressed,  is  strained.  A  stone,  beam,  or  mass  of  metal,  in  a 
building  or  in  a  piece  of  framework,,  if  condensed  or  dilated  in  any 
direction,  or  bent,  twisted,  or  distorted  in  any  way,  is  said  to  ex- 
perience a  strain.  A  ship  is  said  to  'strain'  if,  in  launching,  or 
when  working  in  a  heavy  sea,  the  different  parts  of  it  experience 
relative  motions. 

136.'  If,  when  the  matter  occupying  any  space  is  strained  in  any 
way,  all  pairs  of  points  of  its  substance  which  are  initially  at  equal 
distances  from  one  another  in  parallel  lines  remain  equidistant,  it 
may  be  at  an  altered  distance ;  and  in  parallel  lines,  altered,  it  may 
be,  from  their  initial  direction ;  the  strain  is  said  to  be  homogeneous. 

137.  Hence  if  any  straight  line  be  drawn  through  the  body  in  its 
initial  state,  the  portion  of  the  body  cut  by  it  will  continue  to  be  a 
straight  line  when  the  body  is  homogeneously  strained.  For,  if 
ABC  be  any  such  line,  AB  and  BC,  being  parallel  to  one«line  in  the 
initial,  rem.ain  parallel  to  one  line  in  the  altered  state;  and  therefore 
remain  in  the  same  straight  line  with  one  another.  Thus  it  follows 
that  a  plane  remains  a  plane,  a  parallelogram  a  parallelogram,  and  a 
parallelepiped  a  parallelepiped. 

138.  Hence,  also,  similar  figures,  whether  constituted  by  actual 
portions  of  the  substance,  or  mere  geometrical  surfaces,  or  straight  or 
curved  lines  passing  through  or  joining  certain  portions  or  points  of 
the  substance,  similaily  situated  (i.e.  having  corresponding  parameters 
parallel)  v.hen  altered  according  to  the  altered  condition  of  the  body, 
remain  similar  and  similarly  situated  among  one  another. 

139.  The  lengths  of  parallel  hnes  of  the  body  remain  in  the  .same 
proportion  to  one  another,  and  hence  all  are  altered  in- the  same  pro- 
portion. Hence,  and  from  §  137,  we  infer  that  any  plane  figure 
becomes  altered  to  another  plane  figure  which  is  a  dmiiniihcd  or 
magnified  orthographic  projection  of  the  first  on  some  plane. 

The  elongation  of  the  body  along  any  line  is  the  proportion  which 
the  addition  to  the  distance  between  any  two  points  in  that  line  bears 
to  their  primitive  distance. 

140.  Every  orthogonal  projection  of  an  ellipse  is  an  ellipse  (ri:e 
case  of,  a  circle  being  included).  Hence,  and  from  §  139,  we  teC 
that  an  ellipse  remains  an  ellipse;  and  an  ellipsoid  remains  a  sur- 
face of  which  every  plane  section  is  an  ellipse ;  that  is,  remains  an 
ellipsoid. 


46 


PRELIM  IXAR  y. 


141.  The  ellipsoid  which  any  surface  of  the  body  initially  spheri- 
cal becomes  in  the  altered  condition,  may,  to  avoid  circumlocutions, 

be  c.-lIeJ  the  Strain  El^ij^soiJ. 

142.  In  any  absolutely  unrestricted  homogeneous  strain  there  are 
three  directions  (the  three  principal  axes  of  the  strain  ellipsoid),  at 
right  angles  to  one  another,  which  remain  at  right  angles  to  one 
another  in  the  altered  condition  of  the  body.  Along  one  of  these 
the  elongation  is  greater,  and  along  another  less,  than  along  any 
Other  direction  in  the  body.  Along  the  remaining  one  the  elongation 
is  less  than  in  any  other  line  in  the  plane  of  itself  and  the  first  men- 
tioned, artd  greater  than  along  any  other  line  in  the  plane  of  itself 
and  the  second. 

Ni.yk. — Contraction  is  to  be  reckoned  as  a  negative  elongation:  the 
maximum  elongation  of  the  preceding  enunciation  may  be  a  mini- 
mum contraction:  the  minimum  elongation  may  be  a  maximum 
contraction. 

143.  The  ellipsoid  into  which  a  sphere  becomes  altered  may  be 
an  ellipsoid  of  revolution,  or,  as  it  is  called,  a  spheroid,  prolate,  or 
oblate.  There  is  thus  a  maximum  or  minimum  elongation  along 
the  axis,  and  equal  minimum  or  maximum  elongation  along  all  lines 
perpendicular  to  the  axis. 

Or  it  may  be  a  sphere ;  in  which  case  the  elongations  are  equal  in 
all  directions.  The  eftect  is,  in  this  case,  merely  an  alteration  of 
dimensions  without  change  of  figure  of  any  part. 

144.  The  principal  axes  of  a  strain  are  the  principal  axes  of  the 
ellipsoid  into  which  it  converts  a  sphere.  The  principal  elongations 
Oi  a  strain  are  the  elongations  in  the  direction  of  its  principal  axes. 

145.  ^Vhen  the  positions  of  the  principal  axes,  and  the  magnitudes 
of  the  principal  elongations  of  a  strain  are  given,  the  elongation  of 
any  line  of  the  body,  and  the  alteration  of  angle  between  any  two 
lines,  may  be  obviously  determined  by  a  simple  geometrical  construc- 
tion. 

146.  With  the  same  data  the  alteration  of  angle  between  any  two 

planes  of  the  body  may  also  be  easily  determined,  geometrically, 

147.  Let  the  ellipse  of  the  annexed  diagram  represent  the  section 
of  the  strain  ellij)5oid  through  the  greatest  and  least  principal  axes. 

Let  S'OS,  T'OThc  the  two  diameters  of 
this  ellipse,  wliich  are  equal  to  the  mean 
principal  axis  of  the  ellipsoid.  Every 
plane  through  O,  perpendicular  to  the 
plane  of  the  diagram,  cuts  the  ellipsoid 
in  an  ellipse  of  which  one  principal  axis 
is  the  diameter  in  which  it  cuts  the  ellipse 
of  the  diagram,  and  tlie  other,  the  mean  principal  diameter  of  the 
ellipsoid.     Hence  a  i.>lane  through  cither  SS'  or  TT',  perpendicular. 


}:j::E:.fA7irz. 


to  the  p!ar;e  of  the  diagram,  c'jt=  the 
ih':  f.^0  j.r:r.--.:,il  axes  are  t'./^ah  '.:.■- 


]'!  'n  5n  ellipse  of  v-hiih 


143.     The  c^^v'ceratl^n  of -h-  •::: 
to  ti^e  analysis  of  :he  :..:vt  rr'.'iril  ': 

^*  L^t""6y  a:.-:  (>Z  h-  the  c.-ot- 
rca;':.::.-rn  co:.t:3.ct;orj  respect. ■.•^;ly. 
Let   A    le   ar.y   poir.t    of    t:.e   h 


<9C:  =-  (9C;    ar^^    therefore    6C,  = 


a-?.     K-:rtce  the  tv.-o  trlao-!-;,  COA 
z.:A  C,OA,^:t  eq-al  ar.i  ^.ro.lir. 

H-.r.te  C4  exp-irier^ees  r.o  ah-ratlon  of  ler.^th,  h:t  tikes  :he  altereJ 
poiition  C^  in  the  alter-'i  poiit.en  of  the  :,:^'.'.  h.o'.ii^riv.  if  ■  •; 
rrt--:ore  oo  A'6'  orodoee^.  OA'  ir.t  ^-Z  '  et  .il're- ;-ivhv  to  C'.-? 
an-i  C^f  .  -.ve  fir.^1  toat  ti.e  hrte  CA'  -vo'-r.er.ce^^-o  a. •-■:!::.  .':.  i.ojth, 
b-t  tak-:.  tite  aitere-!  po.::ieri  C  ^i. .  " 

unaiterei  in  ler.;/th.  Pier.ce  ^S  139;  ah  h:.vi  in  the  piaoe  -.^e  ha-,e 
just  f!ra-.vn  reroiin  unaltered  in  l-.r.rth  an  :  in  mutual  inci.r.atitn. 
Similarly  ^ve  s-.-;  that  a.l  lines  in  a  pLne  throurn  CA',  per;  :r.  h.,l:j- 
to  the  plane  of  th-:  ':!iigram,  alterin;:  to  a  :  l-Oj  •hr.^uh  C-_A'_.  ;  ■::- 
pentlioular  to  the  plane  of  the  G;arranj,  rer.;ain  un^-lt-r- J  ;n  ivo.-h  an  J 
in  noutual  inclination. 

149.  The  precipe  character  of  the  strain  ^ve  ha-.e  re-v  un;:er  -"tn- 
sideration  ^^^i^Le  elucidated  by  the  foil  O'-vin^  -—Produce  CO.  a:. :  -.ke 
CC  and  ^C,  re-pectivdy  e  ,u^l  to  DC  :.li  OC,.  Join  CA.  C'A\ 
C\A^,  and  C\A\,  by  plain  an  1  ca::-d  lines  as  in  the  ci:.--  -u. 
Then  we  see  that  the  rhombus  CACA'  ^pl^in  lines)  of  tl.-j  h:  '.■■  ;n 
its  initial  state  becomes  the  rhombus  Ci  A^  C\  A\  (do"e  :,/  i..  the 
altered  condition.  N'ovv  imauine  the  bo-^y  thus  str-.n  .j  •  :■  be 
moved  as  a  ri;;id  body  (i.e.  with  its  state  of  strain  kept  un.h-n^ed) 


48 


PRELIMINARY 


till  Ai  coincides  with  A,  and  C\  with  C\  keeping  all  the  lines  of 
the  diagram  still  in  the  same  plane.  A\Ci  will 
take  a  position  in  CA'  produced,  as  shown  in 
the  new  diagram,  and  the  original  and  the  altered 
parallelogram  will  be  on  the  same  base  A  C,  and 
between  the  same  parallels  AC  and  CA\,  and 
their  other  sides  will  be  equally  inclined  on  the 
two  sides  of  a  perpendicular  to  them.  Hence, 
irrespectively  of  any  rotation,  or  other  absolute 
motion  of  the  body  not  involving  change  of  form. 
or  dimensions,  the  strain  under  conijideration 
may  be  produced  by  holding  fast  and  unaltered  the  plane  of  the 
body  through  AC,  perpendicular  to  the  plane  of  the  diagram,  and 
nvaking  every  plane  parallel  to  it  slide,  keeping  the  same  distance, 
through  a  space  proportional  to  this  distance  (i.  e.  different  planes 
parallel  to  the  fixed  one  slide  through  spaces  proportional  to  their 
distances). 

150.  This  kind  of  strain  is  called  a  simple  shear.  The  plane  of 
a  shear  is  a  plane  perpendicular  to  the  undistorted  planes,  and 
parallel  to  the  lines  of  the  relative  motion.  It  has  (i)  the  properly 
that  one  set  of  parallel  planes  remain  each  unaltered  in  itself;  (2) 

that  another  set  of  parallel  planes  remain 
each  unaltered  in  itself.  This  other  set  is 
got  when  the  first  set  and  the  degree  or 
amount  of  shear  are  given,  thus : — Let 
CC^  be  the  motion  of  one  point  of  one 
plane,  relative  to  a  plane  KL  held  fixed— 
th.e  diagram  being  in  a  plane  of  the  shear. 

C/- /"^^  Bisect  CCj  in  N'.     Draw  NA  perpendicular 

'  ^  to  it.     A  plane  peq^endicular  to  the  plane 

of  the  diagram,  initially  through  AC,  and 
finally  through  ACy^,  remains  unaltered  ia 
its  dimensions. 

151.  One  set  of  parallel  undistorted  planes  and  the  amount  of 
their  relative  parallel  shifting  having  been  given,  we  have  just  seen 
how  to  find  the  other  set.  The  shear  may  be  otherwise  vievred,  and 
considered  as  a  shifting  of  this  second  set  of  parallel  planes,  relative 
to  any  one  of  tliem.  The  amount  of  this  relative  shifting  is  of  course 
equal  to  that  of  tlie  first  set,  relatively  to  one  of  them. 

152.  The  pjrincipal  axes  of  a  shear  are  the  lines  of  maximum 
elongation  and  of  maximum  contraction  respectively.  They  may 
be  found  from  the  preceding  construction  (§  150),  thus: — In  the 
plane  of  the  shear  bisect  the  obtuse  and  acute  angles  between  the 
planes  destined  not  to  become  deformed.  The  former  bisecting  line 
is  the  principal  axis  of  elongation,  and  the  latter  is  the  principal 
axis  of  contraction,  in  their  initial  positions.  The  former  angle 
(obtuse)  becomes  equal  to  the  latter,  its  supplement  (acute),  in  the 


KINEMATICS.  49 

altered  condition  of  the  body,  and  the  

lines  bisecting  the  altered  angles  are  the  y"^'^  """n. 

principal  axes  of  the  strain  in  the  altered    /?/<^ ^^- \s 

body.  /     \>>-^<:^  A 

Otherwise,  taking  a  plane  of  shear  for    l/^..,^-^'\\^^?'^~~~-~^^  \ 

the  plane  of  the  diagram,  let  AB  be  a  i r, — -^^ 

line  in  which  it  is  cut  by  one  of  either  -" 

set  of  parallel  planes  of  no  distortion.  On  any  portion  AB  of  this 
as  diameter,  describe  a  semicircle.  Through  C,  its  middle  point, 
draw,  by  the  preceding  construction,  CD  the  initial,  and  CE  the 
final,  position  of  an  unstretched  line.  Join  DA,  DB,  EA,  EB. 
DA,  DB  are  the  initial,  and  EA,  EB  the  final,  positions  of  the 
principal  axes. 

153.  The  ratio  of  a  shear  is  the  ratio  of  elongation  and  contrac- 
tion of  its  principal  axes.  Thus  if  one  principal  axis  is  elongated 
in  the  ratio  i  :  a,  and  the  other  therefore  (§  148)  contracted  in  the 
ratio  a  ;  I,  a  is  called  the  ratio  of  the  shear.  It  will  be  con\enient 
generally  to  reckon  this  as  the  ratio  of  elongation  ;  that  is  to  say, 
to  make  its  numerical  measure  greater  than  unity. 

In  the  diagram  of  §  152,  the  ratio  oi  DB  to  EB,  or  oi  EA  to  DA, 
is  the  ratio  of  the  shear. 

154.  The  amount  of  a  shear  is  the  amount  of  relative  motion  per 
unit  distance  between  planes  of  no  distortion. 

It  is  easily  proved  that  this  is  equal  to  the  excess  of  the  ratio  of 
the  shear  above  its  reciprocal. 

155.  The  planes  of  no  distortion  in  a  simple  shear  are  clearly  the 
circular  sections  of  the  strain  ellipsoid.  In  the  ellipsoid  of  this 
case,  be  it  remembered,  the  mean  axis  remains  unaltered,  and  is  a 
mean  proportional  between  the  greatest  and  the  least  axis. 

156.  If  we  now  suppose  all  lines  perpendicular  to  the  plane  of  the 
shear  to  be  elongated  or  contracted  in  any  proportion,  without 
altering  lengths  or  angles  in  the  plane  of  the  shear,  and  if,  lastly, 
we  suppose  every  line  in  the  body  to  be  elongated  or  contracted  in 
some  other  fixed  ratio,  we  have  clearly  (§  142)  the  most  general 
possible  kind  of  strain. 

157.  Hence  any  strain  whatever  may  be  viewed  as  compounded 
of  a  uniform  dilatation  in  all  directions,  superimposed  on  a  siuiple 
elongation  in  the  direction  of  one  principal  axis  superimposed  on  a 
simple  shear  in  the  plane  of  the  two  other  principal  a.\es. 

153.  It  is  clear  that  these  three  elementary  component  strains  may 
be  applied  in  any  other  order  as  well  as  that  stated.  Thus,  if  the 
simple  elongation  is  made  first,  the  body  thus  altered  must  get  just 
the  same  shear  in  planes  perpendicular  to  the  line  of  elongation 
as  the  originally  unaltered  body  gets  when  the  order  first  stated  is 
followed.  Or  the  dilatatron  may  be  first,  then  the  elongation,  and 
finally  the  shear,  and  so  on. 


50  PRELIMINARY. 

159.  When  the  axes  of  the  ellipsoid  are  lines  of  the  body  whose 
direction  does  not  change,  the  strain  is  said  to  be  pure,  or  unaccom- 
panied by  rotation.  The  strains  we  have  already  considered  were 
pure  strains  accompanied  by  rotations. 

160.  If  a  body  experience  a  succession  of  strains,  each  unaccom- 
panied by  rotation,  its  resulting  condition  will  generally  be  producible 
by  a  strain  and  a  rotation-.  From  this  follows  the  remarkable  corol- 
lary that  three  pure  strains  produced  one  after  another,  in  any  piece 
of  matter,  each  without  rotation,  may  be  so  adjusted  as  to  leave  the 
body  unstrained,  but  rotated  through  some  angle  about  some  axis. 
We  shall  have,  later,  most  important  and  interesting  applications  to 
fluid  motion,  which  will  be  proved  to  be  instantaneously,  or  differ- 
entially, irrotational ;  but  v/hich  may  result  in  leaving  a  whole  fluid 
mass  merely  turned  round  '"rom  its  primitive  position,  as  if  it  had 
been  a  rigid  body.  [The  following  elementary  geometrical  in- 
vestigation, though  not  bringing  out  a  thoroughly  comprehensive 
view  of  the  subject,  affords  a  rigorous  demonstration  of  the  pro- 
position, by  proving  it  for  a  particular  case. 

Let  us  consider,  as  above  (§  150),  a  simple  shearing  motion.  A 
point  0  being  held  fixed,  suppose  the  matter  of  the  body  in  a  plane, 
cutting  that  of  the  diagram  perpendicularly  in  CD,  to  move  in  this 
plane  from  right  to  left  po,rallel  to  CD ;  and  in  other  planes  parallel 
to  it  let  there  be  motions  proportional  to  their  distances  from  O. 
Consider  first  a  shear  from  P  to  D^;  then  from  /\  on  to  D^;  and 
let  0  be  taken  in  a  line  through  Pi,  perpendicular  to  CD.     During 

the  shear  from  F  to  /", 
a  point  Q  moves  of 
course  to  <2i  through  a 
distance  QQ,  =  PF^. 
Choose  Q  midway  be- 
^  tween  P  and  P^,  so  that 

PiQ^  QP-hP^P.  Now,  as  we  have  seen  above  (§  152),  the 
line  of  the  body,  which  is  the  principal  axis  of  contraction  in  the 
shear  from  Q  to  ^1,  is  OA,  bisecting  the  angle  QOE  at  the  be- 
ginning, and  OA^,  bisecting  QxOE  at  the  end,  of  the  wOiole 
motion  considered.  The  angle  between  these  two  lines  is  half 
the  angle  Q^OQ,  that  is  to  say,  is  equal  to  PiOQ.  Hence, 
if  the  plane  CD  is  rotated  through  an  angle  equal  to  PiOQ,  in 
the  plane  of  the  diagram,  in  the  same  way  as  the  hands  of  a  watch, 
during  the  shear  from  Q  to  Q^,  or,  which  is  the  same  thing,  the 
shear  from  P  X.o  P-^,  this  shear  will  be  effected  without  final  rotation 
of  its  principal  axes.  (Iriiagine  the  diagram  turned  round  till  OA^ 
Jies  along  OA.  The  actual  and  the  newly  imagined  position  of  CD 
will  show  how  this  plane  of  the  body  has  moved  during  such  non- 
rotational  shear.) 

Now,  let  the  second  step,  y,  to  P^,  be  made  so  as  to  complete 
the  whole  shear,  P  to  P^,  which  we  have  proposed  to  consider. 
Such  second  partial  shear  may  be  made  by  the  common  shearing 


KINEMATICS.  51 

process  parallel  to  the  new  position  (imaj;ined  in  the  preceding 
parenthesis)  of  CZ>,  and  to  make  it  also  non-rotational,  as  its 
predecessor  has  been  made,  we,  must  turn  further  round,  in  the 
same  direction,  through  an  angle  equal  to  QiOP^.  Thus  in  these 
two  steps,  each  made  non-rotational,  we  have  turned  the  plane  CD 
round  through  an  angle  equal  to  Q^OQ.  But  now,  we  have  a  whole 
shear  PP. ;  and  to  make  this  as  one  non-rotational  shear,  we  must 
turn  CD  through  an  angle  P^OP  only,  which  is  less  than  Q^OQ  by 
the  excess  of  PiOQ  above  QOP.  ~  Hence  the  resultant  of  the  two 
shears,  PP^,  P^P.^,  each  separately  deprived  of  rotation,  is  a  single 
shear  PP^,  and  a  rotation  of  its  principal  axes,  in  the  direction  of 
the  hands  of  a  watch,  through  an  angle  equal  to  QOP^-POQ. 

161.  Make  the  two  partial  shears  each  non-rotationally.  Return 
from  their  resultant  in  a  single  non-rotational  shear:  we  conclude  with 
the  body  unstrained,  but  turned  through  the  angle  QOP  -POQ,  in 
the  same  direction  as  the  hands  of  a  watch.] 

162.  As  there  can  be  neither  annihilation  nor  generation  of  matter 
in  any  natural  motion  or  action,  the  whole  quantity  of  a  fluid  within 
any  space  at  any  time  must  be  equal  to  the  quantity  originally 'in 
that  space,  increased  by  the  whole  quantity  that  has  entered  it,  and 
diminished  by  the  whole  quantity  that  has  left  it.  This  idea,  when 
expressed  in  a  perfectly  comprehensive  manner  for  every  portion  of 
a  tluid  in  motion,  constitutes  what  is  commonly  called  the  '■equation 
of  continuity.^ 

163.  Two  ways  of  proceeding  to  express  this  idea  present 
themselves,  each  affording  instructive  views  regarding  the  properties 
of  fluids.  In  one  we  consider  a  definite  portion  of  the  fluid ;  follow 
it  in  its  motions;  and  declare  that  the  average  density  of  the  substance 
varies  inversely  as  its  volume.  We  thus  obtain  the  equation  of  con- 
tinuity in  an  integral  form. 

The  form  under  which  the  equation  of  continuity  is  most  commonly 
gi\'en,  or  the  differential  equation  of  continuity,  as  we  may  call  it,  ex- 
presses that  the  rate  of  diminution  of  the  density  bears  to  the  density. 
at  any  instant,  tb.e  same  ratio  as  the  rate  of  increase  of  the  volume  of 
an  infinitely  small  portion  bears  to  the  volume  of  this  portion  at  the 
same  instant. 

164.  To  find  the  differential  equation  of  continuity,  imagine  a 
space  fixed  in  the  interior  of  a  fluid,  and  consider  the  fluid 
which  flows  into  this  space,  and  the  fluid  which  flows  out  of  it, 
across  different  parts  of  its  bounding  surface,  in  any  time.  If  the 
fluid  is  of  the  same  density  and  incompressible,  the  whole  quantity  of 
matter  in  the  space  in  question  must  remain  constant  at  all  times,  and 
therefore  the  quantity  flowing  in  must  be  equal  to  the  quantity  flowing 
out  in  any  time.  If,  on  the  contrary,  during  any  period  of  motion, 
more  fluid  enters  than  leaves  the  fixed  space,  there  ^ill  be  condensa- 
tion of  matter  in  that  space  ;  or  if  more  fluid  leaves  than  enters,  there 
will  be  dilatation.     The  rate  of  augmentation  of  the  average  density 


52  FRELnrmARY. 

of  the  fluid,  per  unit  of  time,  in  the  fixed  space  in  question,  bears  to 
the  actual  density,  at  any  instant,  the  same  ratio  that  the  rate  of 
acquisition  of  matter  into  that  space  bears  to  the  whole  matter  in  that 
space. 

165.  Several  references  have  been  made  in  preceding  sections  to 
the  number  of  independent  variables  in  a  displacem.ent,  or  to  the 
degrees  oi  freedom  or  constraint  under  which  the  displacement  takes 
place.  It  may  be  well,  therefore,  to  take  a  general  (but  cursory)  view 
of  this  part  of  the  subject  itself. 

166.  A  free  point  has  three  degrees  of  freedom,  inasmuch  as  the 
most  general  displacement  which  it  can  take  is  resolvable  into  three, 
parallel  respectively  to  any  three  directions,  and  independent  of  each 
other.  It  is  generally  convenient  to  choose  these  three  directions  of 
resolution  at  right  angles  to  one  another. 

If  the  point  be  constrained  to  remain  always  on  a  given  surface, 
cue  degree  of  constraint  is  introduced,  or  there  are  left  but  two 
degrees  of  freedom..  For  we  may  take  the  normal  to  the  surface 
as  one  of  three  rectangular  directions  of  resolution.  No  displacement 
can  be  effected  parallel  to  it :  and  the  other  two  displacements,  at 
right  angles  to  each  other,  in  the  tangent  plane  to  the  surface,  are 
independent. 

If  the  point  be  constrained  to  remain  on  each  of  two  surfaces,  it 
loses  two  degrees  of  freedom,  and  there  is  left  but  one.  In  fact, 
it  is  constrained  to  remain  on  the  curve  which  is  common  to  both 
surfaces,  and  along  a  curve  there  is  at  each  point  but  one  direction 
of  displacement. 

167.  Taking  next  the  case  of  a  free  rigid  system,  we  have  evidently 
six  degrees  of  freedom  to  consider — three  independent  displacements 
or  translations  in  rectangular  directions  as  a  point  has,  and  three 
independent  rotations  about  three  mutually  rectangular  axes. 

If  it  have  one  point  fixed,  the  system  loses  three  degrees  of  free- 
dom ;  in  fact,  it  has  now  only  the  rotations  above  mentioned. 

This  fixed  point  may  be,  and  in  general  is,  a  point  of  a  continuous 
surface  of  the  body  in  contact  with  a  continuous  fixed  surface.  These 
surfaces  may  be  supposed  *  perfectly  rough,'  so  that  sliding  may  be 
impossible. 

If  a  second  point  be  fixed,  the  body  loses  two  more  degrees  of 
freedom,  and  keeps  only  one  freedom  to  rotate  about  the  line  joining 
the  two  fixed  points. 

If  a  third  point,  not  in  a  line  with  the  other  two,  be  fixed,  the  body 
is  fixed, 

168.  If  one  point  of  the  rigid  system  is  forced  to  remain  on  a 
smooth  surfiice,  one  degree  of  freedom  is  lost ;  there  rems.m  Jive,  two 
displacements  in  the  tangent  plane  to  the  surface,  and  three  rotations. 
As  an  additional  degree  of  freedom  is  lost  by  eacli  successive  limita- 
tion of  a  point  in  the  body  to  a  smooth  surface,  six  such  conditions 
completely  determine  the  position  of  the  body.    Thus  if  six  points 


KINEMATICS,  S3 

property  chosen  t*n  the  barrel  and  stock  of  a  rifle  be  made  to  rest  on 
six  convex  portions  of  the  surface  of  a  fixed  rigid  body,  the  rifle  may 
be  replaced  any  number  of  times  in  precisely  the  same  position,  for 
the  purpose  of  testing  its  accuracy. 

A  fixed  V  under  the  barrel  near  the  muzzle,  and  another  under 
the  swell  of  the  .stock  close  in  front  of  the  trigger-guard,  give  four 
of  the  contacts,  bearing  the  weight  of  the  rifle.  A  fifth  (the  one 
to  be  broken  by  the  recoil)  is  supplied  by  a  nearly  vertical  fixed 
plane  close  behind  the  second  V,  to  be  touched  by  the  trigger-guard, 
the  rifle  being  pressed  forward  in  its  V's  as  far  as  this  obstruction 
allows  it  to  go.  This  contact  may  be  dispensed  with  and  nothing 
sensible  of  accuracy  lost,  by  having  a  mark  on  the  second  V,  and  a 
corresponding  mark  on  barrel  or  stock,  and  sliding  the  barrel  back- 
wards or  forwards  in  the  V's  till  the  two  marks  are,  as  nearly  as 
can  be  judged  by  eye,  in  the  same  plane  perpendicular  to  the  barrel's 
axis.  The  sixth  contact  may  be  dispensed  with  by  adjusting  two 
marks  on  the  heel  and  toe  of  the  butt  to  be  as  nearly  as  need  be 
in  one  vertical  plane  judged  by  aid  of  a  plummet.  This  method 
requires  less  of  costly  apparatus,  and  is  no  doubt  more  accurate  and 
trustworthy,  and  more  quickly  -and  easily  executed,  than  the  ordi- 
nary method  of  clamping  the  rifle  in  a  massive  metal  cradle  set  on  a 
heavy. mechanical  slide. 

A  geometrical  clamp  is  a  means  of  applying  and  maintaining 
six  mutual  pressures  between  tv/o  bodies  touching  one  another  at 
six  points. 

A  'geometrical  slide'  is  any  arrangement  to  apply  five  degrees 
of  constraint,  and  leave  one  degree  of  freedom,  to  the  relative  motion 
of  two  rigid  bodies  by  keeping  them  pressed  together  at  just  live 
points  of  their  surfaces. 

Ex.  I.  The  transit  instrument  would  be  an  instance  if  one  end 
of  one  pivot,  made  slightly  convex,  were  pressed  a^Ljainst  a  fixed 
vertical  end-plate,  by  a  spring  pushing  at  the  other  end  of  the  axis, 
The  other  four  guiding  points  are  the  points,  or  small  areas,  of  con- 
tact of  the  pivots  on  the  Y's. 

Ex.  2.  Let  two  rounded  ends  of  legs  of  a  three  legged  stool 
rest  in  a  straight,  smooth,  V-shaped  canal,  and  the  third  on  a  smooth 
horizontal  plane.  Gravity  maintains  positive  determinate  pressures 
on  the  five  bearing  points ;  and  there  is  a  determinate  distribution 
and  amount  of  friction  to  be  overcome,  to  produce  the  rectilineal 
translational  motion  thus  accurately  provided  for. 

Ex.  3.  Let  only  one  of  the  feet  rest  in  a  V  canal,  and  let 
another  rest  in  a  trihedral  hollow  in  line  with  the  canal,  the  third 
still  resting  on  a  horizontal  plane.  There  are  thus  six  bearing  points, 
one  on  the  horizontal  plane,  tv/o  on  the  sides  of  the  canal,  and  three 
on  the  sides  of  the  trihedral  hollov; :  and  the  stool  is  fixed  in  a 
determinate  position  as  long  as  all  these  six  contacts  are  unbroken. 
Substitute  for  gravity  a  spring,  or  a  screw  and  nut  (of  not  infinitely 
rigid  material),  binding  the  stool  to  the  rigid  body  to  which  these- 
six  planes  belong.     Thus  we  have  a    'geometrical  clamp,'  which 


54  PRELIMINARY. 

clamps  two  bodies  together  with  perfect  firmness  m  a  perfectly 
definite  position,  without  the  aid  of  friction  (except  in  the  screw, 
If  a  screw  is  used) ;  and  in  various  practical  applications  gives 
very  readily  and  conveniently  a  more  securely  fiml  connexion  by 
one  screw  slightly  pressed,  than  a  clamp  such  as  those  commonly 
made  hitherto  by  mechanicians  can  give  with  three  strong  screws 
forced  to  the  utmost. 

Do  away  with  the  canal  and  let  two  feet  (instead  of  only  one)  rest 
on  the  plane,  the  other  still  resting  in  the  conical  hollow.  The 
number  of  contacts  is  thus  reduced  to  five  (three  in  the  hollow 
and  two  on  the  plane),  and  instead  of  a  'clamp'  we  have  again 
a  slide.  This  form  of  slide, — a  three-legged  stool  with  two  feet 
resting  on  a  plane  and  one  in  a  hollow, — will  be  found  very  useful  in 
a  large  variety  of  applications,  in  which  motion  about  an  axis  is  de- 
sired when  a  material  axis  is  not  conveniently  attainable.  Its  first 
application  was  to  the  '  azimuth  mirror,'  an  instrument  placed  on 
the  glass  cover  of  a  mariner's  compass  and  used  for  taking  azimuths 
of  sun  or  stars  to  correct  the  compass,  or  of  landmarks  or  other 
terrestrial  objects  to  find  the  ship's  position.  It  has  also  been  applied 
to  the  '  Deflector,'  an  adjustible  magnet  laid  on  the  glass  of  the 
compass  bowl  and  used,  according  to  a  principle  first  we  believe 
given  by  Sir  Edward  Sabine,  to  discover  the  'semicircular'  eiTor 
produced  by  the  ship's  iron.  The  movement  may  be  made  very 
frictionless  when  the  plane  is'  horizontal,  by  weighting  the  move- 
able body  so  that  its  centre  of  gravity  is  very  nearly  over  the 
foot  that  rests  in  the  hollow.  One  or  two  guard  feet,  not  to  touch 
the  plane  except  in  case  of  accident,  ought  to  be  added  to  give 
a  broad  enough  base  for  safety. 

The  geometrical  slide  and  the  geometrical  clamp  have  both  been 
found  very  useful  in  electrometers,  in  the  '  siphon  recorder,'  and  in 
an  instrument  recently  brought  into  use  for  automatic  signalling 
through  submarine  cables.  An  infinite  variety  of  forms  may  be 
given  to  the  geometrical  slide  to  suit  varieties  of  application  of  the 
general  principle  on  which  its  definition  is  founded. 

An  old  form  of  the  geometrical  clamp,  with  the  six  pressures  pro- 
duced by  gravity,  is  the  three  V  grooves  on  a  stone  slab  bearing  the 
three  legs  of  an  astronomical  or  magnetic  instrument.  It  is  not 
generally  however  so  'well-conditioned'  as  the  trihedral  hole,  the 
V  groove,  and  the  horizontal  plane  contact,  described  above. 

There  is  much  room  for  improvement  by  the  introduction  of 
geometrical  slides  and  geometrical  clamps,  in  the  mechanism  of 
mathematical,  optical,  geodetic,  and  astronomical  instruments: 
which  as  made  at  present  are  remarkable  for  disregard  of  geome- 
trical and  dynamical  principles  in  their  slides,  micrometer  screws, 
and  clamps.  Good  workmanship  cannot  compensate  for  bad  design, 
whether  in  the  safety-valve  of  an  ironclad,  or  the  movements  and 
adjustments  of  a  theodolite. 

169.  If  one  point  be  constrained  to  remain  in  a  curve,  there  remain 
four  degrees  of  freedom. 


KINEMATICS.  55 

If  two  points  be  constrained  to  remain  in  given  curves,  there  are 
our  degrees  of  constraint,  and  we  have  left  two  degrees  of  freedom. 
Dne  of  these  may  be  regarded  as  being  a  simple  rotation  about  the 
ine  joining  the  constrained  parts,  a  motion  which,  it  is  clear,  the 
)ody  is  free  to  receive.  It  may  be  shown  that  the  other  possible 
■notion  is  of  the  most  general  character  for  one  degree  of  freedom  ; 
.hat  is  to  say,  translation  and  rotation  in  any  fixed  proportions,  as  of 
he  nut  of  a  screw. 

If  one  line  of  a  rigid  system  be  constrained  to  remain  parallel  to 
.tself,  as  for  instance,  if  the  body  be  a  three-legged  stool  standing  on 
I  perfectly  smooth  board  fixed  to  a  common  window,  sliding  in  its 
frame  v.-ith  perfect  freedom,  there  remain  three,  displacements  and  one 
rotation. 

But^we  need  not  farther  pursue  this-  subject,  as  the  number  of 
combinations  that  might  be  considered  is  almost  endless ;  and  those 
already  given  sufrice  to  show  how  simple  is  the  determination  of  the 
degrees  of  freedom  or  constraint  in  any  case  that  may  present  itself. 

170.  One  degree  of  constraint  of  the  most  general  cliaracter,  is  not 
producible  by  constraining  one  point  of  the  body  to  a  curve  surf.Tce ; 
but  it  consists  in. stopping  one  line  of  the  body  from  longitudinal 
motion;  except  accompanied  by  rotation  round  this  line,  in  fixed 
pro{)ortion  to  the  longitudinal  motion.  E\ery  other  motion  being 
left  unimpeded,  there  remains  free  rotation  about  any  axis  perpen- 
dicular to  that  line  (two  degrees  of  freedom);  and  translation  in  any 
direction  perpendicular  to  tlie  same  line  (two  degrees  of  ireedom). 
These  last  four,  v/ith  the  one  degree  of  freedom  to  screw,  con- 
stitute the  five  degrees  of  freedom,  v,-hich,  v.'ith  one  degree  oi  con- 
straint, make  up  the  six  elements.  This  condition  is  realized  in  the 
following  mechanical  an-angement,  which  seems  the  sinipiest  that 
can  be  imagined  for  the  purpose  : — 

Let  a  screw  be  cut  on  one  shaft  of  a  Hooke's  joint,  and  let  the 
other  shaft  be  joined  to  a  fixed  shaft  by  a  secojid  Hooke's  joint. 
A  nut  turning  on  that  screw-shoft  has  the  most  general  kind  of 
motion  admitied  v.men  there  is  one  degree  of  cons;ttaint.  Or  it  is 
subjected  to  just  one  degree  of  constraint  of  the  most  general  clia- 
racter. It  has  five  degrees  of  freedom  ;  for  it  may  mcive,  ist,  by 
screwing  on  its  shaft,  the  two  Flooke's  joints  being  at  rest;  2\\'\, 
it  may  rotate  about  either  axis  of  tlie  first  Hooke's  joinf,  cr  any  ax;3 
in  their  plime  (two  more  degrees  of  freedom  :  bein.g  tVeedom  to  rorue 
about  two  axes  through  one  point) ;  3rd,  it  may,  by  the  two  Hooke's 
joints,  each  bending,  have  translation  without  rotation  in  any  direction 
perpendicular  to  the  link  or  shaft  between  tlie  two  Hooke's  joints 
(two  more  degrees  of  freedom).  But  it  cannot  have  a  motion  of 
translation  parallel  to  the  line  of  the  link  without  a  definite  propor- 
tion of  rotation  round  this  line ;  nor  can  it  have  rotation  round  this 
line  without  a  definite  proportion  of  translation  parallel  to  it. 


CHAPTER   11, 

DYNAMICAL  LAWS  AND  PRINCIPLES. 


17L  In  the  preceding  chapter  we  considered  as  a  subject  of  pure 
geometry  the  motion  of  points,  lines,  surfaces, -and  volumes,  whetlier 
taking  place  with  or  without  change  of  dimensions  and  form;  and  the 
results  we  there  arrived  at  are  of  course  altogether  independent  of  the 
idea  of  matter,  and  of  \he.  forces  which  matter  exerts.  We  have  here- 
tofore assumed  the  existence  merely  of  motion,  distortion,  etc.;  we 
now  come  to  the  consideration,  not  of  how  we  might  consider  such 
motion,  etc,  to  be  produced,  but  of  the  actual  causes  which  in  the 
material  woHd  do  produce  them.  The  axioms  of  the  present  chapter 
must  therefore  be  considered  to  be  due.  to  actual  experience,  in  the 
shape  either  of  observation  or  experiment.  How  such  experience  is 
to  be  conducted  will  form  the  subject  of  a  subsequent  chapter. 

172.  We  cannot  do  better,,  at  all  events  in  commencing,  than  follow 
Newton  somewhat  closely.  Indeed  the  introduction  to  the  Frincipia 
contains' in  a  most  lucid  form  the  general  foundations  of  dynamics. 
The  Definitiones  and  Axiomatq,  sive  Leges  MotHs,  there  laid  down, 
require  only  a  few  amplifications  and  additional  illustrations,  suggested 
by  subsequent  developments,  to  suit  them  to  the  present  state  of 
science,  and  to  make  a  much  better  introduction  to  dynamics  than 
we  find  in  even  some  of  the  best  modern  treatises. 

173.  We  cannot,  of  course,  give  a  definition  of  Matter  which  will 
satisfy  the  metaphysician ;  but  the  naturalist  may  be  content  to  know 
matter  as  that  which  can  be  perceived  by  the  sefises,  or  as  that  ivhich  can 
be  acted  upon  by,  or  can  exert,  force.  The  latter,  and  indeed  the 
former  also,  of  these  definitions  involves  the  idea  of  Force,  which,  in 
point  of  fact,  is  a  direct  object  of  sense;  probably  of  all  our  senses, 
and  certainly  of  the  '  muscular  sense.'  To  our  chapter  on  Properties 
of  Matter  we  must  refer  for  further  discussion  of  the  question,  What 
is  matter  i 

174.  The  Quantity  of  Matter  in  a  body,  or,  as  we  now  call  it,  the 
Mass  of  a  body,  is  proportional,  according  to  Newton,  to  the  Volume 
and  the  Density  conjointly.  In  reality,,  the  definition  gives _us^ the 
meaning  of  density  rather  than  of  mass ;  for  it  shows  us  that  if  twice 
the  original  quantity  of  matter,  air  for  example,  be  lorced  into  a  vessel 


DyyA.VICAL  LAJVS  AXD  PRL\'CIPLES.  57 

of  given  capacity,  the  density  will  be  doubled,  and  so  on.  But  it  also 
shows  us  that,  of  matter  of  uniform  density,  the  mass  or  quantity  is 
proportional  to  the  volume  or  space  it  occupies. 

Let  Af  be  the  mass,  p  the  density,  and  V  the  volume,  of  a  homo 
tjeneous  body.     Then 

M=  Vp; 

if  we  so  take  our  units  that  unit  of  mass  is  that  of  unit  volume  01  a 
body  of  unit  density. 

If  the  density  be  not  uniform,  the  equation 

M=-  Vp 

gives  the  Average  (§  26)  density;  or,  as  it  is  usually  called,  the  M'can 
density,  of  the  body. 

It  is  worthy  of  particular  notice  that,  in  this  definition,  Xcwton 
says,  if  there  be  anything  which  /nv/r  pervades  the  interstices  of  all 
bodies,  this  is  fiot  taken  account  of  in  estimating  their  Mass  or 
Density. 

175.  Xowton  further  states,  that  a  practical  measure  of  the  mass 
of  a  body  is  its  Weight.  His  experiments  on  pendulums,  by  wliich  he 
establishes  this  most  important  ri^mark,  v,-ill  be  described  later,  in  our 
chapter  on  Properties  of  Matter. 

As  will  be  presently  explained,  the  unit  mass  most  convenient  for 
British  measurem.ents  is  an  imperial  pound  of  matter. 

176.- The  Qua ; it :ty  of  Motion,  or  the  AfoiHintmn,  of  a  rigitl  body 
moving  v.'itliout  rotation  is  proportional  to  its  mass  and  vciLOcity  con- 
jointly. The  whole  motion  is  the  sum  of  the  motions  of  its  several 
parts.  Thus  a  doubled  mass,  or  a  doubled  velocity,  would  correspond 
to  a  double  quantity  of  m.otion  ;  and  so  on. 

Hence,  if  we  take  as  unit  of  momentum  the  momentum  of  a  unit 
of  matter  moving  with  unit  velocity,  the  momentum  of  a  n^.r.ss  M 
moving  with  velocity  v  is  Mv. 

177.  Chiifige  of  Qihuitity  of  M>ti:.'\  or  Cha^i.y  of  Mo'ncntam,  is 
proportional  to  the  mass  movir.g  and  tlie  change  of  its  VL-locity 
conjointly. 

Change  of  velocity  is  to  be  understood  in  the  general  sense  0^  ^  31. 
Thus,  in  tlie  figure  of  that  section,  if  a  velocity  rcpivjcntcd  by  OA  he 
changed  to  another  represented  by  OC^  the  cb.ange  of  velocity  is 
represented  in  magnitude  and  direction  by  A  C. 

178.  Rate  of  Cliange  of  Momciitum,  or  Accoloration  of  Monuntum,  is 

proportional  to  the  mass  moving  and  the  acceleration  of  its  velocity 

conjointly.     Thus  (§  44)   the   rate   of  ch.aage   of  momentum  of  a 

falling  body  is  constant,  and  in  the  vertical  direction.     Again  (^  iG) 

the  rate  of  change  of  momentum  of  a  mass  M,  describing  a  circle  of 

1/  V^ 
radms  R,  with  uniform  velocity  F,  is  — /,—  ,  and  is  directed  to  the 

A 

centre  of  the  circle;  that  is  to  say, ^it  depends  upon  a  change  of  di 

lection,  not  a  change  of  speed,  of' the  motion. 


S8  PRELIMINARY. 

179.  The  Vis  Viva,  or  Kinetic  Energy,  of  a  moving  body  is  pro- 
portional  to  the  mass  and  the  square  of  the  velocity,  conjointly.  If 
we  adopt  the  same  units  of  mass  and  velocity  as  before,  there  is 
particular  advantage  in  defining  kinetic  energy  as  hal/iht  product  of 
the  mass  and  the  square  of  its  velocity. 

180.  Rate  of  Change  of  Kinetic  Energy  (when  defined  as  above)  is 
the  product  of  the  velocity  into  the  component  of  acceleration  of 
momentum  in  the  direction  of  motion. 

Suppose  the  velocity  of  a  mass  J/  to  be  changed  from  v  to  v^  in 
any  time  t;  the  rate  at  which  the  kinetic  energy  has  changed  is 

t^ov^  ~M{v  -v)  is  the  rate  o^  change  of  momentum  in  the  dir^^c- 

T 

tion  of  motion,  and  J  {v,  +  v)  is  equal  to  v,  if  t  be  infinitely  small. 
Hence  the  above  statement.  It  is  often  convenient  to  use  Newton's 
Fluxional  notation  for  the  rate  of  change  of  any  quantity  per  unit  of 

time.     In  this  notation  (§  28)  ^  stands  for  -  {v  -v) ;  so  that  the  rate 

T 

of  change  oi^Mv^,  the  kinetic  energy,  is  Mv.  v.  (See  also  §§229,  241.) 

181.  It  is  to  be  observed  that,  in  what  precedes,  with  the  exception  of 
the  definition  of  density,  we  have  taken  no  account  of  the  dimensions, 
of  the  moving  body.  This  is  of  no  consequence  so  long  as  it  does 
not  rotate,  and  so  long  as  its  parts  preserve  the  same  relative  positions 
amongst  one  another.  In  this  case  we  may  suppose  the  whole  of  the 
matter  in  it  to  be  condensed  in  one  point  or  particle*.  We  thus  speak 
of  a  material  particle,  as  distinguished  from  d.  geometrical  point.  If  the 
body  rotate,  or  if  its  parts  change  their  relative  positions,  then  we 
cannot  choose  any  one  point  by  whose  motions  alone  we  may  de- 
termine those  of  the  other  points.  In  such  cases  ^he  momentum  and 
change  of  mom.entum  of  the  whole  body  in  any  direction  are,  the 
sums  of  the  momenta,  and  of  the  changes  of  momentum,  of  its  parts, 
in  ^hese  directions;  while  the  kinetic  energy  of  the  v/hole,  being  non- 
directional,  is  simply  the  sum  of  the  kinetic  energies  of  the  several 
parts  or  particles,  • 

182.  Matter  has  an  innate  power  of  resisting  external  influences, 
so  that  every  body,  so  far  as  it  can,  remains  at  rest,  or  moves  uni- 
formly in  a  straight  line. 

This,  the  Inertia  of  matter,  is  proportional  to  the  quantity  of  matter 
in  the  body.  And  it  follows  that  some  caiise  is  requisite  to  disturb  a 
body's  uniformity  of  motion,  or  to  change  its  direction  from  the 
natural  rectilinear  path, 

183.  Force  is  any  cause  which  tends  to  alter  a  body's  natural  state 
of  rest,  or  of  uniform  motion  in  a  straight  line. 

Force  is  wholly  expended  in  the  Action  it  produces ;  and  the  body, 
after  the  force  ceases  to  act,  retains  by  its  inertia  the  direction  of 


DYNAAflCAL  LAWS  AND  PRIMCIPLES.  59 

motion  and  the  velocity  which  were  given  to  it  Force  may  be  of 
divers  kinds,  as  pressure,  or  gravity,  or  friction,  or  any  of  the  attractive 
or  repulsive  actions  of  electricity,  magnetism,  etc. 

184.  The  three  elements  specifying  a  force,  or  the  three  elements 
which  must  be  known,  before  a  clelr  notion  of  the  force  under  con- 
sideration can  be  formed,  are,  its  place  of  application,  its  direction, 
and  its  magnitude. 

(a)  The  place  of  application  of  a  force.  The  first  case  to  be  con- 
sidered is  that  in  which  the  place  of  application  13  a  point.  It  has 
'been  shown  already  in  what  sense  the  term  'point'  is  to  be  taken, 
and,  therefore,  in  what  way  a  force  may  be  imagined  as  acting  at  a 
point.  In  reality,  however,  the  place  of  application  of  a  force  is 
always  either  a  surface  or  a  space  of  three  dimensions  occupied  by 
matter.  The  point  of  the  finest  needle,  or  the  edge  of  the  sharpest 
knife,  is  still  a  surface,  and  acts  as  such  on  the  bodies  to  which  it 
may  be  applied.  Even. the  most  rigid  substances,  when  brought 
together,  do  not  touch  at  a  point  merely,  but  mould  each  other  so 
as  to  produce  a  surface  of  application.  On  the  other  hand,  gravity' 
is  a  force  of  which  the  place  of  application  is  the  whole  matter  of  the 
body  whose  weight  is  considered ;  and  the  smallest  particle  of  matter 
that  has  weight  occupies  some  finite  portion  of  space.  Thus  it  is  to 
be  remarked,  that  there  are  two  kinds  of  force,  distinguishable  by 
their  place  of  application — force  whose  place  of  application  is  a 
surface,  and  force  whose  place  of  application  is  a  solid.  When  a 
heavy  body  rests  on  the  ground,  or  on  a  table,  force  of  the- second 
character,  acting  downwards,  is  balanced  by  force  of  the  first  character 
acting  upwards. 

ib)  The  second  element  in  the  specification  oi  a  force  is  its 
direction.  The  direction  of  a  force  is  the  line  in  which  it  acts. 
If  the  place  of  application  of  a  force  be  regarded  as  a  point,  a 
line  through  that  point,  in  the  direction  in  which  the  force  tends  to 
move  the  body,  is  the  direction  of  the  force.  In  the  case  of  a  force 
distributed  over  a  surface,  it  is  frequently  possible  and  convenient 
to  assume  a  single  point  and  a  single  line,  such  that  a  certain  force 
acting  at  tliat  point  in  that  line  would  produce  the  same  effect  as  is 
really  produced. 

{c)  The  third  element  in  the  specification  of  a  force  is  its  magnitude. 
This  involves  a  consideration  of  the  method  followed  in  dynamics  for 
measuring  forces.  Before  measuring  an}l;hing  it  is  necessar)-  to  have 
a  unit  of  measurement,  or  a  standard  to  which  to  refer,  and  a  prin- 
ciple of  numerical  specification,  or  a  mode  of  referring  to  the  standard. 
These  will  be  suppljed  presently.     See  also  §  224,  below.. 

185.  The  Measure  of  a  Force  is  the  quantity  of  motion  which  it 
produces  in  unit  of  time. 

The  reader,  who  has  been  accustomed  to  speak  of  a  force  of  so 
many  pounds,  or  so  many  tons,  may  be  reasonably  startled  when  he 
finds  that  Newton  gives  no  countenance  to  such  expressions.  Tlie 
method  is  not  correct  unless  it  be  specified  at  what  part  of  the  earth's 


<5o  PRELIMINARY. 

surface  the  pound,  or  other  definite  quantity  of  matter  named,  is  to 
be  weighed ;  for  the  weight  of  a  given  quantity  of  matter  differs  in 
different  latitudes. 

It  is  often,  however,  convenient  to  use  instead  of  the  absolute 
unit  (§  i88),  the  gravitation  unit — which  is  simply  the  weight  of  unit 
mass.  It  must,  of  course,  be  specified  in  what  latitude  the  observation 
is  made.  Thus,  let  W  be  the  mass  of  a  body  in  pounds ;  g  the 
velocity  it  would  acquire  in  falling  for  a  second  under  the  influence 
of  its  weight,  or  the  earth's  attraction  diminished  by  centrifugal 
force ;  and  F  its  weight  measured  in  kinetic  or  absolute  units.  We 
have  p=  Wg. 

The  force  of  gravity  on  the  body,  in  gravitation  units,  is  W. 

186.  According  to  the  system  commonly  followed  in  mathe* 
matical  treatises  on  dynamics  till  fourteen  years  ago,  when  a  small 
instalment  of  the  first  edition  of  the  present  work  was  issued  for 
the  use  of  our  students,  the  unit  of  mass  was  g  times  the  mass  of 
the  standard  or  unit  weight.  This  definition,  giving  a  varying  and  a 
very  unnatural  unit  of  mass,  was  exceedingly  inconvenient.  By  taking 
the  gravity  of  a  constant  mass  for  the  unit  of  force  it  makes  the  unit 
of  force  greater  in  high  than  in  low  latitudes.  In  reality,  standards 
of  weight  are  7nasses,  not  forces.  They  are  employed  primarily  in 
commerce  for  the  purpose  of  measuring  out  a  definite  qimntity  of 
matter ;  not  an  amount  of  matter  which  shall  be  attracted  by  the 
earth  with  a  given  force. 

A  merchant,  with  a  balance  and  a  set  of  standard  weights,  v/ould 
give  his  customers  the  same  quantity  of  the  same  kind  of  matter 
however  the  earth's  attraction  might  vary,  depending  as  he  does  upon 
weii^hts  for  his  measurement ;  another,  using  a  spring-balance,  would  , 
defraud  his  customers  in  high  latitudes,  and  himself  in  low,  if  his 
instrument  (which  depends  on  constant  forces  and  not  on  the  gravity 
of  constant  masses)  were  correctly  adjusted  in  London. 

It  is  a  secondary  application  of  our  standards  of  weight  to  employ 
them  for  the  measurement  oi  forces,  such  as  steam  pressures,  mus- 
cular pov.'cr,  etc.  In  all  cases  where  great  accuracy  is  required, 
the  results  obtained  by  such  a  method  have  to  be  reduced  to 
what  they  would  have  been  if  the  measurements  of  force  had  been 
made  by  means  of  a  perfect  spring-balance,  graduated  so  as  to 
indicate  the  forcv-s  of  gravity  on  the  standard  weights  in  some  con- 
ventional localit}'. 

It  is  therefore  very  much  simpler  and  better  to  take  the  imperial 
pound,  or  other  national  or  international  standard  weight,  as,  for 
instance,  the  gramme  (see  the  chapter  on  Measures  and  Instru- 
ments), as  the  unit  of  mass,  and  to  derive  from  it,  according  to 
Newton's,  definition  above,  the  unit  of  force.  This  is  the  method 
which  Gauss  has  adopted  in  his  great  improvement  of  the  system  of 
measurement  of  forces. 

187.  The  formula,  deduced  by  Clairault  from  observation,  and  a 
certain  theory  regarding  the  figure  and  density  of  the  earth,  may  be 


DYNAMICAL  LAWS  AND  PR/NCIPLES.  6i 

employed  to  calculate  the  most  probable  value  of  the  apparent  force 
of  gravity,  being  the  resultant  of  true  gravitation  and  centrifugal  force, 
in  any  locality  where  no  pendulum  observation  of  sufficient  accuracy 
has  been  made.  This  formula,  with  the  two  coefficients  which  it 
involves,  corrected  according  to  modern  pendulum  observations,  i* 
as  follows  : — 

Let  G  be  the  apparent  force  of  gravity  on  a  unit  mass  at  the 
equator,  and ^  that  in  any  latitude  A.;  then 

^=  (?(i  +  '00513  sin'X). 

The  value  of  Cr,  in  terms  of  the  absolute  unit,  to  be  explained 
iimmediately,  is 

32'oS8. 

According  to  this  formula,  therefore,  polar  gravity  will  be 
^=32-088  X  roo5i3  =  32'252. 

188.  As  gravity  does  not  furnish  a  definite  standard,  independent 
of  locality,  recourse  must  be  had  to  something  else.  The  principle 
of  measurement  indicated  as  above  by  Xewton,  but  first  introduced 
practically  by  Gauss  in  connexion  with  national  standard  masses, 
furnishes  us  with  what  we  want.  According  to  this  principle,  the 
standard  or  unit  force  is  that  force  ^ohic/i,  acting  on  a  national  standard 
unit  of  matter  during  the  unit  of  time,  generates  the  unit  of  velocity. 

This  is  known  as  Gauss'  absolute  unit ;  absolute,  because  it  fur- 
nishes a  standard  force  independent  of  the  differing  amounts  of 
gravity  at  different  localities.  It  is  however  terrestrial  and  incon- 
stant if  the  unit  of  time  depends  on  the  earth's  rotation,  as  it  does 
in  our  present  system  of  chronometr)'.  The  period  of  vibration  of 
a  piece  of  quartz  crystal  of  specified  shape  and  size  and  at  a  stated 
temperature  (a  tuning-fork,  or  bar,  as  one  of  the  bars  of  glass  used 
in  the  'musical  glasses 'j  gives  us  a  unit  of  time  which  is  constaiit 
through  all  space  and  all  time,  and  independent  of  the  earth.  A 
unit  of  force  founded  on  such  a  unit  of  time  would  be  better  entitled 
to  the  designation  absolute  than  is  the  '  absolute  unit '  now  generally 
adopted,  which  is  founded  on  the  mea)'.  solar  second.  But  this  de- 
pends essentially  on  one  particular  piece  of  mailer,  and  is  therefore 
liable  to  all  the  accidents,  etc.  v.hich  affect:  so-called  National 
Standards  however  carefully  they  may  be  preserved,  as  well  as  to- 
the  almost  insuperable  practical  difficulties  which  are  experienced 
when  we  attempt  to  make  exact  copies  of  them.  Stiil,  in  the  present 
state  of  science,  we  are  really  confined  to  such  approximations.  I'iie 
recent  discoveries  due  to  the  Kinetic  theory  of  gases  and  to  Si)ectrun> 
analysis  (especially  when  it  is  applied  to  the  light  of  the  hea\enly 
bodies)  indicate  to  us  natural  standard  «^:)ieces  oi  matter  such  as 
atoms  of  hydrogen,  or  sodium,  ready  maLle  in  infinite  numbers,  all 
absolutely  alike  in  every  physical  projjerty.  The  time  of  vibrati-ju 
of  a  sodium  particle  corresponding  to  any  one  of  its  inuaes  of  v.Lra- 
tion,  is  known  to  be  absolutely  independent  of  its  po:^iti'j;i  ::i  tl^e 
universe,  and  it  will  probably  remain  the  same  so  long  as  tl'.e  particle 


62  PRELIMINARY. 

itself  exists.  The  wave-length  for  that  particular  ray,  i.e.  the  space 
through  which  light  is  propagated  in  vacuo  during  the  time  of  one 
complete  vibration  of  this  period,  gives  a  perfectly  invariable  unit  of 
length  ;  and  it  is  possible  that  at  some  not  very  distant  day  the  mass 
of  such  a  sodium  particle  may  be  employed  as  a  natural  standard  for 
the  remaining  fundamental  unit.  This,  the  latest  im.provement  made 
upon  our  original  suggestion  of  a  Feretmial  Spring,  is  due  to  Clerk 
Maxwell. 

189.  The  absolute  unit  depends  on  the  unit  of  matter,  the  unit  of 
time,  and  the  unit  of  velocity;  and  as  the  unit  of  velocity  depends  on 
the  unit  of  space  ana  the  unit  of  time,  there  is,  in  the  definition,  a 
single  reference  to  mass  and  space,  but  a  ^i?//^'/^' reference  to  time; 
and  this  is  a  pjint  that  must  be  particularly  attended  to. 

190.  The  unit  of  mass  may  be  the  British  imperial  pound,  or> 
better,  the  gramme;  the  unit  of  space  the,  British  standard  foot,  or, 
better,  the  centimetre  ;  and  the  unit  of  time  the  mean  solar  second. 

We  accordingly  define  the  British  absolute  unit  force  as  '  the  force 
which,  acting  on  one  pound  of  matter  for  one  second,  generates  a 
velocity  of  one  foot  per  second.' 

191.  To  render  this  standard  intelligible,  all  that  has  to  be  done  is 
to  find  how  many  absolute  units  will  produce,  in  any  particular  locality, 
the  same  effect  as  the  force  of  gravity  on  a  given  mass.  The  way  to 
do  this  is  to  measure  the  effect  of  gravity  in  producing  acceleration 
on  a  body  unresisted  in  any  way.  The  most  accurate  method  is 
indirect,  by  means  of  the  pendulum.  The  result  of  pendulum  ex- 
perinlents  made  at  Leith  Fort,  by  Captain  Kater,  is,  that  the  velocity 
acquired  by  a  body  falling  unresisted  for  one  second  is  at  that  place 
32'207  feet  per  second.  The  preceding  formula  gives  exactly  32-2, 
for  the  latitude  55°  35',  which  is  appro'ximately  that  of  Edinburgh. 
The  variation  in  the  force  of  gravity  for  one  degree  of  difference  of 
latitude  about  the  latitude  of  Edinburgh  is  only  '0000832  of  its  own 
amount.  It  is  nearly  the  same,  though  somewhat  mq^e,  for  every 
degree  of  latitude  southwards,  as  far  as  the  southern  limits  of  the 
British  Isles.  On  the  other  hand,  the  variation  per  degree  would  be 
sensibly  less,  as  far  north  as  the  Orkney  and  Shetland  Isles.  Hence 
the  augmentation  of  gravity  per  degree  from  south  to  north  through- 
out the  British  Isles  is  at  most  about  x^h^rr)  ^^  '^^  whole  amount  in 
any  locality.  The  average  for  the  whole  of  Great  Britain  and  Ireland 
differs  certainly  but  little  from  32*2.  Our  present  application  is,  that 
the  force  of  gravity  at  Edinburgh  is  32-2  times  the  force  which,  acting 
on  a  pound  for  a  second,  would  generate  a  velocity  of  one  foot  per 
second;  in  other  words,  32-2  is  the  number  of  absolute  units  which 
measures  the  weight  of  a  pound  in  this  latitude.  Thus,  speaking 
very  roughly,  the  British  absolute  unit  of  force  is  equal  to  the  weight 
of  about  half  an  ounce. 

192.  Forces  (since  they  involve  only  direction  and  magnitude)  may 
be  represented,  as  velocities  are,  by  straight  lines  in  their  directions, 
and  of  lengths  proportional  to  their  magnitudes,  respectively. 


DY^^AMICAL  LAWS  AND  PRINCIPLES.  b^ 

Also  the  laws  of  composition  and  resolution  of  any  nun:iber  of 
forces  acting  at  the  same  point,  are,  as  we  slrall  show  later  (§  221), 
the  same  as  those  which  we  have  already  jirovc  J  to  hold  for  velo 
cities;  'sO  that  witii  the  substitution  of  force  for  \clocity,  §§  30,  31 
are  still  true. 

193.  .  The  Compiv:;tit  of  a  force  in  any  direction,  somctinius 
called  the  Effective  Component  in  that  direction,  is  therefore  found 
by  multiplying  the  magnitude  of  the  force  by  tiie  cosine  of  the 
angle  between  the  directions  of  tlie  force  and  ti^.c  component.  The 
remaining  component  in  this  case,  is  perpendicular  to  the  other. 

It  is  Very  generally  convenient  to  resolve  forces  into  components 
parallel  to  tiiree  lines  at  right  angles  to  each  other;  each  such  reso- 
lution being  effected  by  multiplying  by  the  cosine  of  the  angle 
concerned. 

194.  [If  any  number  of  points  be  placed  in  any  positions  in  space, 
another  can  be  found,  such  that  its  distance  from  any  plane  what- 
ever is  the  mean  of  their  distances  from  that  plane;  and  df  one  or 
more  of  the  given  points  be  in  motion,  the  velocity  of  the  mean 
point  perpendicular  to  the  plane  is  the  mean  of  the  velocities  of 
the  others  in  the  same  direction. 

If  we  take  two  points  A-^,  A.,  the  middle  point,  /*,,  of  the  line 
joining  them  is  obviously  distant  from  any  plane  whatever  by  a 
quantity  equal  to  the  mean  (in  this  case  the  half  sum  or  difference 
as  they  are  on  the  same  or  on  opposite  sides)  of  their  distances 
from  that  plane.  Hence  twice  the  distance  of  /*,  from  any  plane 
is  equal  to  the  (algebraic)  sum  of  the  distances  of  A^,  A  from  it. 
Introducing  a  third  point  A^^,  if  we  join  A^P^  and  divide  it  in  P^ 
so  that  AjP3=  2P3P2,  three  times  the  distance  of  P^  from  any  plane 
is  equal  to  the  sura  of  the  distance  of  A3  and  twice  that  of  P^  from 
the  same  plane:  i.e.  to  the  sum  of  the  distances  of  A^,  A.-.,  and  A^ 
from  it;  or  irs  distance  is  the  mean  oi  theirs.  And  so  on  for  any 
number  of  points.  Th©  proof  is  exceedingly  simple.  Thus  suppose 
P^  to  be  the  ?nt\i'i  of  the  first  //  i^oin'o  A-^,  .-t.,...A,^:  and  A^^^  any 
other  point.      Divide  A^^^P^  in  P.^,   so  that  A^^^P^^^  ~  tiP^^^p^. 

Then    from     P^,     P\^,     A^^^y    draw     pcrj^-e:> 

diculars  to  any  plane,  Ujccting  it  in  S,    7,    K 

Draw  P^QP  parallel  to  STF.     Then 

Hence    n  +  i(2P^^^  =  PA_^^^.     Add    to     these 

n+  I C? 7"  and  its  equal  nP^_S+  RV,  and  wc  get      5-   'j' 

«^i((?A^,  +  Q.T)  -  nP^S  T  R  ;%  A'.-/,.,,, 

i.e.    ^r^iP„,^T=nP^S-^A^^,K 

In  words,  «  +  1  times  liie  distance  of  P^^^  from  any  plane  is  ei]aal 
to  that  of  ^,^j  with  n  tunes  that  of  P^,  i.e.  equal  to  the  saiii  of  liie 


64  PRELIMINAR  Y. 

distances  of  A^y  A^,...A,^^^  from  the  plane.  Thus  if  the  proposition 
be  true  for  any  number  of  points,  it  is  true  for  one  more — and  so  on 
— but  it  is  obviously  true  for  two,  hence  for  three,  and  therefore 
generally.  And  it  is  obvious  that  the  order  in  which  the  points  are 
taken  is  immaterial. 

As  the  distance  of  this  point  from  any  plane  is  the  mean  of  the 
distances  of  the  given  ones,  the  rate  of  increase  of  that  distance, 
i.e.  the  velocity  perpendicular  to  the  plane,  must  be  the  mean  of  the 
rates  of  increase  of  their  distances — i.e.  the  mean  of  their  velocities 
perpendicular  to  the  plane.] 

195.  The  Centre  of  Inertia  or  Mass  of  a  system  of  equal  material 
points  (whether  connected  with  one  another  or  not)  is  the  point 
whose  distance  is  equal  to  their  average  distance  from  any  plane 
whatever  (§  194). 

A  group  of  material  points  of  unequal  masses  may  always  be 
imagined  as  composed  of  a  greater  number  of  equal  material  points, 
because  we  may  imagine  the  given  material  points  divided  into  dif- 
ferent numbers  of  ver}'  small  parts.  In  any  case  in  which  the  magni- 
tudes of  the  given  masses  are  incommensurable,  we  may  approach  as 
near  as  we  please  to  a  rigorous  fulfilment  of  the  preceding  statement, 
by  making  the  parts  into  which  we  divide  them  sufficiently  small. 

On  this  understanding  the  preceding  definition  may  be  applied 
to  define  the  centre  of  inertia  of  a  system  of  material  points,  whether 
given  equal  or  not.     The  result  is  equivalent  to  this  : — • 

The  centre  of  inertia  of  any  system  of  material  points  whatever 
(whether  rigidly  connected  with  one  another,  or  connected  in  any 
way,  or  quite  detached),  is  a  point  whose  distance  from  any  plane 
is  equal  to  the  sum  of  the  products  of  each  mass  into  its  distance 
from  the  same  plane  divided  by  the  sum  of  the  masses. 

We  also  see,  from  the  proposition  stated  above,  that  a  point  whose 
distance  from  three  rectangular  planes  fulfils  this  condition,  must 
fulfil  this  condition  also  for  every  other  plane. 

The  co-ordinates  of  the  centre  of  inertia,  of  masses  Wj,  «'o,  etc., 
at  points  (.Ti,  }\,  z^,  {x^^  y.,,  z^,  etc.,  are  given  by  the  following 
formulae : — 

7''';.'.-:.  -f-  iv.p:.-^  +  etc.      ln'X     _     "^.^vy     _     Sr.'S 

These  formulae  are  perfectly  general,  and  can  easily  be  put  into 
the  particular  shape  required  for  any  given  case. 

The  Centre  of  Inertia  or  Mass  is  thus  a  perfectly  definite  point  in 
every  body,  or  group  of  bodies.  The  term  Centre  of  Gravity  is  often 
very  inconveniently  used  for  it.  The  theory  of  the  resultant  action  of 
gravity,  which  will  be  given  under  Abstract  Dynamics,  shows  that, 
except  in  a  definite  class  of  distributions  of  matter,  there  is  no  fixed 
point  which  can  properly  be  called  the  Centre  of  Gravity  of  a  rigid 
body.  In  ordinary  cases  of  terrestrial  gravitation,  however,  an  ap- 
proximate solution  is  available,  according  to  which,  in  common  par- 
lance, the  term   Ce?itre  of  Gravity  may  be  used  as  equivalent  to 


DYXA.mCAL  LAWS  AND  PRINCIPLES.  65 

Centre  of  Inertia ;  but  it  must  be  carefully  remembered  that  the  fun- 
damental ideas  involved  in  the  two  definitions  are  essentially  different. 
The  second  proposition  in  §  194  may  now  evidently  be  stated 
thus : — The  sum  of  the  momenta  of  the  parts  of  the  system  in  any 
direction  is  equal  to  the  momentum  in  the  same  direction  of  a  mass 
equal  to  the  sum  of  the  masses  mgving  with  a  velocity  equal  to  the 
velocity  of  the  centre  of  inertia. 

196.  The- mean  of  t"he  squares  of  the  distances  of  the  centre  of 

p    inertia,  /,  from  each  of  the  points  of  a  system 

^-r^    is  less  tlian  the  mean  of  the, squares  of  the  dis- 

^^^^/   \      tance  of  any  other  point,  O,  from  them  by  the 

^o^^    /  square  of  01.     Hence  the  centre  of  inertia  is 

-.;:_ y i     the  point   the   sum    of  the  squares   of  whose 

^  V  u    distances  from  any  given  points  is  a  minimum. 

For  OP  ^01-  -^  IP^  +  2OI IQ.  P  being  any  one  of  the  points 
and  PQ  perpendicular  to  01.  But  IQ  is  the  distance  of  P  from 
a  plane  through  /  perpendicular  to  0Q_.  Hence  the  mean  of  all 
distances,  IQ,  is  zero.     Hence, 

(mean  oi  IP'')  =  (mean  of  OP^)  -  OP,  which  is  the  proposition. 

197.  Again,  the  mean  of  the  squares  of  the  distances  of  the  points 
of  the  system  from  any  line,  exceeds  the  corresponding  quantity  for 
a  parallel  line  through  the  centre  of  inertia,  by  the  square  of  the 
distance  between  these  lines. 

For  in  the  above  figure,  let  the  plane  of  the  paper  represent  a 
plane  tjuough  /  perpendicular  to  these  lines,  0  the  point  in  v/hich 
the  first  line  meets  it,  P  the  point  in  which  it  is  met  by  a  parallel 
line  through  any  one  of  the  points  of  the  system.  Draw,  as  before, 
PQ  perpendicular  to  01.  Then  PI  is  the  perpendicular  distance, 
from  tiie  axis  through  /,  of  the  point  of  the  system  considered,  PO 
is  its  distance  from  the  first  axis,  01  the  distance  between  the  two 
axes. 

Then,  as  before, 

(mean  of  OP')  =  OP  4-  (mean  of  IP'); 
since  the  mean  of  IQ   is   still   zero,  IQ  being  the   distance  of  a 
point  of  the  system  from  the  plane  through  /perj^endicular  to  01. 

198.  If  the  masses  of  the  points  be  unequal,  it  is  easy  to  see  (as 
in  ^  195)  that  the  first  of  these  theorems  becomes— 

The  sum  of  the  squares  of  the  distances  of  the  parts  of  a  system 
from  any  point,  each  multiplied  by  the  mass  of  that  part,  exceeds  the 
corresponding  quantity  for  the  centre  of  inertia  by  the  product  of 
the  square  of  tivj  distance  of  the  point  from  the  centre  of  inertia,  by 
the  whole  mass  of  the  system. 

Also,  the  sum  of  the  products  of  the  mass  of  each  part  of 
a  system  by  the  square  of  its  distance  from  any  axis  is  calicd  the 
Moment  0/  Inertia  of  the  system  about  this  axis;  and  the  second 
proposition  above  is  equivahnt  to — 

Vui.  -j:;— t 


66  preliminary: 

The  moment  of  inertia  of  a  system  about  any  axis  is  equal  to  the 
moment  of  inertia  about  a  parallel  axis  through  the  centre  of  inertia, 
/,  together  with  the  moment  of  inertia,  about  the  first  axis,  of  the 
whole  mass  supposed  condensed  at  /. 

199.  The  Moment  of  any  physical  agency  is  the  numerical  mea- 
sure of  its  importance.  Thus,  the  moment  of  inertia  of  a  body 
round  an  axis  (§  198)  means  the  im.portance  of  its  inerna  relatively 
to  rotation  round  that  axis.  Again,  the  moment  of  a  force  round  a 
pomt  or  round  a  line  (§  46),  signifies  the  measure  of  its  importance  as 
regards  producing  or  balancing  rotation  round  that  point  or  round 
that  line. 

It  is  often  convenient  to  represent  the  moment  of  a  force  by  a  line 
numerically  equal  to  it,  drawn  through  the  vertex  of  the  triangle 
representing  its  magnitude,  perpendicular  to  its  plane,  through  the 
front  of  a  watch  held  in  the  plane  with  its  centre  at  the  point,  and 
facing  so  that  the  force  tends  to  turn  round  this  point  in  a  direction 
opposite  to  the  hands.  The  moment  of  a  force  round  any  axis  is  the 
moment  of  its  component  in  any  plane  perpendicular  to  the  axis, 
round  the  point  in  which  the  plane  is  cut  by  the  axis.  Here  we 
imagine  the  force  resolved  into  two  components,  one  parallel  to  the 
axis,  which  is  ineffective  so  far  as  rotation  round  the  axis  is  con- 
cerned; the  other  perpendicular  to  the  axis  (that  is  to  say,  having  its 
line  in  any  plane  perpendicular  to  the  a>ds).  This  latter  component 
may  be  called  the  effecdve  component  of  the  force,  with  reference 
to  rotation  round  the  axis.  And  its  moment  round  the  axis  may  be 
defined  as  its  moment  round  the  nearest  point  of  the  axis,  which  is 
equivalent  to  the  preceding  definition..  It  is  clear  that  the  moment 
of  a  force  round  any  axis,  is  equal  to  the  area  of  the  projection  on 
any  plane  perpendicular  to  the  axis,  of  the  figure  representing  its 
moment  round  any  point  of  the  axis. 

200.  [The.  projection  of  an  area,  plane  or  curved,  on  aiiy  plane, 
is  the  area  included  in  the  projection  of  its  bounding  line. 

If  we  imagine  an  area  divided  into  any  number  of  parts,  the  pro- 
jections of  these  parts  on  any  plane  make  up  the  projection  of  the 
whole.  But  in  this  statement  it  must  be  understood  that  the  areas 
of  partial  projections  are  to  be  reckoned  as  positive  if  particular 
sides,  which,  for  brevity,  we  may  call  the  outside  of  the  projected 
area  and  the  front  of  the  plane  of  projection,  face  the  same  way, 
and  negative  if  they  face  oppositely. 

Of  course  if  the  projected  surface,  or  any  part  of  it,  be  a  plane  area 
at  right  angles  to  the  plane  of  projection,  the  projection  vanishes. 
The  projections  of  any  two  shells  having  a  common  edge,  on  any 
plane,  are  equal.  The  projection  of  a  closed  surface  (or  a  shell  with 
evanescent  edge),  on  any  plane,  is  nothing. 

Equal  areas  in  one  plane,  or  in  parallel  planes,  have  equal  projec- 
tions on  any  plane,  whatever  may  be  their  figures. 

Hence  the  i)rojection  of  any  plane  figure,  or  of  any  shell  edged 
by  a  plane  figure,  on  another  plane,  is  equal  to  its  area,  multiplied 


DYNAMICAL   LAWS  AND  PRLNCIPLES.  67 

by  the  cosine  of  the  angle  at  which  its  plane  is  inclined  to  the  plane 
of  projection.  This  angle  is  acute  or  obtuse,  according  as  the  out- 
side of  the  projected  area,  and  the  front  of  the  plane  of  projection, 
face  on  the'  whole  towards  the  same  parts,  or  oppositely.  Hence 
lines  representing,  as  above  described,  moments  about  a  point  in 
different  planes,  are  to  be  compounded  as  forces  are.  See  an 
analogous  theorem  in  §  107.] 

201.  A  Couple  is  a  pair  of  equal  forces  acting  in  dissimilar  direc- 
tions in  parallel  lines.  The  Moment  of  a  couple  is  the  sum  of  the 
moments  of  its  forces  about  any  point  in  their  plane,  and  is  therefore 
equal  to  the  product  of  either  force  into  the  shortest  distance  between 
their  directions.     This  distance  is  called  the  Arm  of  the  couple. 

The  Axis  of  a  Couple  is  a  line  drawTi  from  any  chosen  point  of 
reference  perpendicular  to  the  plane  of  the  couple,  of  such  magnitude 
and  in  such  direction  as  to  represent  the  magnitude  of  the  moment, 
and  to  indicate  the  direction  in  which  the  couple  tends  to  turn.  The 
most  convenient  rule  for  fulfilling  the  latter  condition  is  this: — Hold 
a  watch  with  its  centre  at  the  point  of  reference,  and  with  its  plane 
parallel  to  the  plane  of  the  couple.  Then,  according  as  the  motion 
of  the  hands  is  contrary  to,  or  along  with  the  direction  in  which  the 
couple  tends  to  turn,  draw  the  a.xis  of  the  couple  through  the  face  or 
through  tlie  back  of  the  watch.  It  will  be  found  that  a  couple  is 
completely  represented  by  its  axis,  and  that  couples  are  to  be  resolved 
and  compounded  by  the  same  geometrical  constructions  performed 
with  reference  to  their  axes  as  forces  or  velocities,  with  reference  to 
the  lines  directly  representing  them. 

202.  By  introducing  in  the  definition  of  moment  of  velocity  (§  46) 
the  mass  of  the  moving  body  as  a  factor,  we  have  an  important 
element  of  dynamical  science,  the  Moment  of  Momentum.  Tlie 
laws  of  composition  and  resolution  are  the  same  as  those  already 
explained. 

203.  [If  the  point  of  application  of  a  force  be  displaced  through 
a  small  space,  the  resolved  part  of  the  displacement  in  the  direcdon 
of  the  force  has  been  called  its  Virtual  Vil.slfy.  Tiiis  is  positive  or 
negative  according  as  the  virtual  velocity  is  in  the  same,  or  in  the 
opposite,  direction  to  that  of  the  force. 

The  product  of  the  force,  into  the  virtual  velocity  of  its  point  of 
apphcation,  has  been  called  the  Virtual  Moment  of  the  force.  These 
terms  we  have  introduced  since  they  stand  in  the  history  and  develop- 
ments of  the  science  ;  but,  as  we  shall  show  further  on,  they  are 
inferior  su'ostltutes  for  a  far  more  useful  set  of  ideas  clearly  laid  down 
by  Newton.] 

204.  A  force  is  said  to  do  work  if  its  place  of  application  has  a 
positive  componont  motion  in' its  direction  ;  and  the  work  done  Ly  it 
is  measured  by  the  product  of  its  amount  into  this  component  motion. 

Generally,  unit  of  work  is  done  by  unit  force  acting  througli  unit 
space.     In  lifung  coals  from  a   pit,  the   amount  of  work  done    is 


68  PRELIMINARY. 

proportional,  to  the  weight  of  the  coals  lifted ;  that  is,  to  the  force 
overcome  in  raising  them  ;  and  also  to  the  height  through  which  they 
are  raised  The  unit  for  the  measurement  of  work  adopted  in  practice 
by  British  engineers,  is  that  required  to  overcome  a  force  equal  to  the 
weight  of  a  pound  through  the  space  of  a  foot ;  and  is  called  a  Foot- 
Pound.     (See  §185.) 

In  purely  scientific  measurements,  the  unit  of  work  is  not  the  foot- 
pound, but  the  kinetic  unit  force  (§  190)  acting  through  unit  of  space. 
Thus,  for  example,  as  we  shall  show  further  on,  this  unit  is  adopted 
in  measuring  the  work  done  by  an  electric  current,  the  units  for 
electric  and  magnetic  measurements  being  founded  upon  the  kinetic 
unit  force. 

If  the  weight  be  raised  obliquely,  as,  for  instarice,  along  a  smooth 
inclined  plane,  the  space  through  which  the  force  has  to  be  overcome 
is  increased  in  the  ratio  of  the  length  to  the  height  of  the  plane  ;  but 
the  force  to  be  overcome  is  not  the  whole  weight,  but  only  the  resolved 
part  of  the  weight  parallel  to  the  plane ;  and  this  is  less  than  the 
weight  in  the  ratio  of  the  height  of  the  plane  to  its  length.  By  multi- 
plying these  two  expressions  together,  we  find,  as  we  might  expect, 
that  the  amount  of  work  requhed  is  unchanged  by  the  substitution  of 
the  oblique  for  the  vertical  path. 

205.  Generally,  for  any  force,  the  work  done  during  aji  indefinitely 
small  displacement  of  the  point  of  application  is  the  virtual  moment 
of  the  force  (§  203),  or  is  the  product  of  the  resolved  part  of  the  force 
in  the  direction  of  the  displacement  into  the  displacement. 

From  this  it  appears,  that  if  the  motion  of  the  point  of  application 
be  always  perpendicular  to  the  direction  in  v/hich  a  force  acts,  such  a 
force  does  no  work.  Thus  the  mutual  normal  pressure  between  a 
/ixed  and  moving  body,  the  tension  of  the  cord  to  which  a  pendulum 
bob  is  attached,  or  the  attraction  of  the  sun  on  a  planet  if  the  planet 
describe  a  circle  with  the  sun  in  the  centre,  are  all  instances  in  which 
no  work  is  done  by  the  force. 

205.  The  work  done  by  a  force,  or  by  a  couple,  upon  a  body 
turning  about  an  axis,  is  the  product  of  the  moment  of  either  into  the 
angle  (in  circular  measure)  through  which  the  body  acted  on  turns,  if 
the  moment  remains  the  same  in  all  positions  of  the  body.  If  the 
moment  be  variable,  the  above  assertion  is  only  true  for  indefinitely 
small  displacements,  but  maybe  made  accurate  by  employing  the  proper 
average  moment  of  the  force  or  of  the  couple.    The  proof  is  obvious. 

207.  Work  done  on  a  body  by  a  force  is  always  shown  by  a  cor- 
resjjonding  increase  of  vis  viva,  or  kinetic  energy,  if  no  other  forces 
act  on  the  body  which  can  do  work  or  have  work  done  against  them. 
If  work  be  done  against  any  forces,  the  increase  of  kinetic  energy  is 
less  than  in  the  former  case  by  the  amount  of  work  so  done.  In 
virtue  of  this,  however,  the  body  possesses  an  ecjuivalent  in  the  form 
of  Pofent'nl  E'hrgy  (^  239),  if  its  physical  conditions  are  such  that 
tliese  forces  will  act  equally,  and  in  the  same  directions,  if  the  motion 
of  the  system  is  reversed.     Thus  there  may  be  no  change  of  kinetic 


DYNAMICAL  LAWS  AND  FRI.XCIPLES.  69 

energy  produced,  and  the  work  done  may  be  wholly  stored  tip  as 
potential  energy. 

Thus  a  weight  requires  work  to  raise  it  to  a  height,  a  spring  requires 
work  to  bend  it,  air  requires  work  to  compress  it,  e;c.  ;  but  a  raised 
weight,  a  bent  spring,  compressed  air,  etc.,  are  stores  of  energy  which 
can  be  made  use  of  at  pleasure. 

208.  In  what  precedes  we  have  given  some  of  Newton's  Drfinitiones 
nearly  in  his  own  words  ;  others  have  been  enunciated  in  a  form  more 
suitable  to  modem  methods ;  and  some  terms  have  been  introduced 
which  were  invented  subsequent  to  the  publication  of  the  Pruicipia. 
But  the  Axiomata,  sive  Leges  Motus,  to  which  we  now  proceed,  are 
given  in  Newton's  own  words.'  The  two  centuries  which  have  nearly 
elapsed  since  he  first  gave  them  have  not  shown  a  necessity  for  any 
addition  or  modification.  The  first  two,  indeed,  were  discovered  by 
Gahleo  :  and  the  third,  in  some  of  its  many  forms,  was  known  to 
Hooke,  Huyghens,  Wallis,  Wren,  and  others,  before  the  publication 
of  the  Frincipia.  Of  late  there  has  been  a  tendency  to  di'.ide  the. 
second  law  into  two,  called  respectively  the  second  and  third,  and  to 
ignore  the  third  entirely,  though  using  it  directly  in  ever)'  dynamical 
problem ;  but  ail  who  have  done  so  have  been  forced  indirectly  to 
acknowledge  the  incompleteness  of  their  substitute  forNewton's  system, 
by  introducing  as  an  axiom  what  is  called  D' Alembert's  principle,  which 
is  really  a  deduction  from  Newton's  rejected  third  law.  Newton's  ov.-n 
interpretation  of  his  third  law  directly  points  out  not  only  D'Alembert's 
principle,  but  also  the  modern  principles  of  Work  and  Energy. 

209.  An  Axiom  is  a  proposition,  the  truth  of  which  must  be  ad- 
mitted as  soon  as  t!ie  terms  in  which  it  is  expressed  are  cleaily 
understood.  And,  cs  v,-e  shal!  show  in  our  ch.apter  on  '  Experience,' 
physical  axioms  are  axiomatic  to  tiiose  v.-ho  have  sufncient  knowiecge' 
of  physical  phenomena  to  enable  them  to  understand  perfectly  what 
is  asserted  by  them.  Without  further  remark  we  shail  gi'.'c  Newton's 
Three  L;tw5 ;  it  being  remembered  that,  as  the  properties  of  matter 
might  have  been  such  as  to  rendi.r  a  totally  differLnt  bct  of  laws 
axiomatic,  these  laws  must  be  considered  as  resting. on.  convictions 
drawn  from  observation  and  experiment,  riot  on  intuitise  perception. 

210.  Lex  I,  Corpus  omnc  pcrsr.-er.u-e  in  statu  su?  i]uir::endi  ~,(l 
movendi  uui/ormitcr  in  directum,  jjisi  guater.us  liiud  a  viribus  :?"pressis 
ccgitur  statum  suum  mutare. 

Every  body  contirrues  in  its  state  of  rest  or  of  ur'-form  v::tion  in  a 
straifit  line,  exce/'t  m  so  far  as  it  may  be  compclUd  by  impri.ssed  forces 
to  change  that  state. 

211.  The  meaning  of  the  term  Rest,  in  physical  science,  cannot  be 
a!.)solutely  defineil,  in  ismuch  as  absolute  rest  nowhere  exists  in  n„t.ire. 
If  the  universe  of  matter  were  finite,  its  centre  of  inertia  miglit  f.i'rl}- 
be  considered  as  absolutely  at  rest;  or  it  might  be  imagine. 1  tj  bo 
moving  with  any  uniform  velocity  m  any  direction  wh:itever  tlnou,;;: 
infinite  space.     But  it  is  remarkable  that   the    first    law  of   iwjVojw 


70 


PRELIMINARY. 


enables  us  (§  215,  below)  to  explain  what  may  be  called  directionat 
rest.  Also,  a5  will  b6  seen  farther  on,  a  perfectly  smooth  spherical 
body;  'made  up  of  concentric  shells,  each  of  uniform  material  and 
density  throughout,  if  made  to  revolve  about  an  axis,  will,  in  spite  oj 
impressed  forces  f  revolve  with  uniform  angular  velocity,  and  will  main- 
tain its  axis  of  revolution  in  an  absolutely  fixed  direction.  Or,  as  will 
soon  be  shown  (§  233),  the  plane  in  which  the  moment  of  momentuni 
of  the  universe  (if  finite)  round  its  centre  of  inertia  is  the  greatest, 
which  is  clearly  determinable  from  the  actual  motions  at  any  instant, 
is  fixed  in  direction  in  space. 

212.  We  may  logically  convert  the  assertion  of  the  first  law  of 
motion  as  to  velocity  into  the  following  statements ; — 

The  times  during  which  any  particular  body,  not  compelled  by 
force  to  alter  the  speed  of  its  motion,  passes  through  equal  spaces, 
are  equal.  And,  again—- Every  other  body  in  the  universe,  not  com- 
pelled by  force  to  alter  the  speed  of  its  motion,  moves  over  equal 
spaces  in  successive  intervals,  during  which  the  particular  chosen  body 
moves  over  equal  spaces. 

213.  The  first  part  merely  expresses  the  convention  universally 
adopted  for  the  measurement  of  Time,  The  earth  in  its  rotation 
about  its  axis,  presents  us  with  a  case  of  motion  in  which  the  con* 
dition  ot  not  being  compelled  by  force  to  alter  its  speed,  is  more 
weaxly  fulfilled  than  in  any  other  which  we  can  easily  or  accurately 
observe.  And  the  numerical  measurement  of  time  practically  rests 
on  defining  eqiml  intervals  of  time,  as  times  during  which  the  earth  turns 
through  equal  angles.  This  is,  of  course,  a  mere  convention,  and 
not  a  law  of  nature ;  and,  as  we  now  see  it,  is  a  part  of  Newton's 
first  law. 

214.  The  remainder  of  the  law  is  not  a  convention,  but  a  great 
truth  of  nature,  which  we  may  illustrate  by  referring  to  small  and 
trivial  cases  as  well  as  to  the  grandest  phenomena  we  can  conceive. 

A  curling-stone,  projected  along  a  horizontal  surface  of  ice,  travels 
equal  distances,  except  in  so  far  as  it  is  retarded  by  friction  and  by 
the  resistance  of  the  air,  in  successive  inten-als  of  time  during  which 
the  earth  turns  through  equal  angles.  The  sun  moves  through  equal 
portions  of  interstellar  space  in  times  during  which  the  earth  turns 
through  equal  angles,  except  in  so  far  as  the  resistance  of  interstellar 
matter,  and  the  attraction  of  other  bodies  in  the  universe,  alter  his 
speed  and  that  of  the  earth's  rotation. 

215.  If  tvvo  material  points  be  projected  from  one  position,  A,  at 
the  same  instant  with  any  velocities  in  any  directions,  and  each  left  to 
move  uninfluenced  by  force,  the  line  joining  them  will  be  always 
parallel  to  a  fixed  direction.  For  the  law  asserts,  as  we  have  seen, 
that  AF  :  AF'  ::  AQ:  AQ,  if/',  Q,  and  again  F\  Q',  are  simulta- 
neous positions;  and  therefore  FQ  is  parallel  to  F'Q'.  Hence  if  four 
material  points  O,  F,  Q,  R  are  all  projected  at  one  instant  from  one 
position,  OF,  OQ,  OR  are  fixed  directions  of  reference  ever  after. 


D  YNAMICAL  LA  WS  AND  PRINCIPLES.  7 1 

But,  practically,  the  determination  of  fixed  directions  in  space 
(§  233)  is  made  to  depend  upon  the  rotation  of  groups  of  particles 
exerting  forces  on  each  other,  and  thus  involves  the  Third  Law  of 
Motion. 

216.  The  whole  law  is  singularly  at  variance  with  the  tenets  of  the 
ancient  philosophers,  who  maintained  that  circular  motion  is  perfect. 

The  last  clause,  ^nist  guaienus,'  etc.,  admirably  prepares  for  the 
introduction  of  the  second  law,  by  conveying  the  idea  that  it  is  force 
done  li'hich  can  produce  a  change  cf  ?notion.  How,  ^^■e  naturally  in- 
quire, does  the  change  of  motion  produced  depend  on  the  m:\gn:L'.:de 
and  direction  of  the  force  which  produces  it?    The  answer  is  — 

217.  Lex  IL  Mutationem  motfcs  proportionahm  esse  zi  inotrid  :;•:- 
prcssae,  et fieri  secundum  lineam  reciam  qua  vis  iUa  impritr.itur. 

Change  of  motion  is  proportional  to  the  impressed  fore:,  and  takes  place 
in  the  direction  of  the  straight  line  in  which  the  force  acts. 

218.  If  any  force  generates  motion,  a  do'ab'e  force  will  generate 
double  motion,  and  so  on,  whether  simuIt:.ncous!y  or  successively, 
instantaneously  or  gradually,  applied.  And  this  motion,  if  the  body 
was  moving  beforehand,  is  either  added  to  the  previous  motion  if 
directly  conspiring  v.'ith  it;  or  is  subtracted  if  directly  opposed;  or 
is  geometrically  compounded  with  it,  according  to  the  kinematical 
principles  already  explained,  if  the  line  of  previous  motion  and  the 
direction  of  the  force  are  inclined  to  each  other  at  any  angle.  (This 
is  a  paraphrase  of  Newton's  own  comments  on  the  second  lav\-.) 

219.  In  Chapter  I.  we  have  considered  change  of  velocit}-,  or 
acceleration,  as  a  purely  geometrical  element,  and  have  seen  how  it 
may  be  at  once  inferred  from  the  given  initial  and  final  velocities  of  a 
body.  By  the  definition  of  a  quantity  of  motion  (3  2i\),  we  see  that, 
if  we  multiply  the  change  of  velocity,  thus  geometrically  determined, 
by  the  mass  of  the  body,  we  have  the  change  of  motion  referred  to  in 
Newton's  law  as  the  measure  of  the  force  which  produces  it. 

It  is  to  be  particularly  noticed,  that  in  this  statement  there  is  nothing 
said  about  the  actual  motion  of  the  body  before  it  was  acted  on  by  the 
force :  it  is  only  the  change  of  motion  that  concerns  us.  Thus  the 
same  force  will  produce  precisely  the  same  change  of  motion  in  a 
body,  whether  the  body  be  at  rest,  or  in  motion  with  any  velocity 
whatever. 

220.  Again,  it  is  to  be  noticed  that  nothing  is  said  as  to  the  body 
being  under  the  action  of  one  force  only ;  so  that  we  may  logically 
put  a  part  of  the  second  law  in  the  following  (apparently)  amplified 
form  : — 

When  any  forces  tuhatever  act  on  a  body,  then,  whether  the  body  be 
origpia/ly  at  rest  or  niainng  with  any  velocity  and  in  any  direction,  each 
force  produces  in  the  body  the  exact  change  of  motion  which  it  zvould  have 
produced  if  it  had  acted  singly  on  the  body  originally  at  rest, 

221.  A  remarkable  consequence  follows  immediately  from  this  view 
of  the  second  law.     Sincig  forces  are  measured  by  the  changes  of 


72  PRELIMINARY. 

motion  they  produce,  and  their  directions  assigned  by  the  directions 
in  which  these  changes  are  produced;  and  since  the  changes  of 
motion  of  one  and  the  same  body  are  in  the  directions  of,  and  pro- 
portional to,  the  changes  of  velocity — a  single  force,  measured  by  the 
resultant  change  of  velocity,  and  in  its  direction,  will  be  the  equivalent 
of  any  number  of  simultaneously  acting  forces.     Hence 

The  resultant  of  any  fiumber  of  forces  {applied  at  one  point)  is  to  be 
found  by  the  same  gcotnctrical  process  as  the  resultant  of  any  jiuniber  of 
sinndta7ieous  velocities. 

222.  From  this  follows  at  once  (§  31)  the  construction  of  the 
Parallelogratn  of  Forces  for  finding  the  resultant  of  two  forces,  and 
the  Polygon  of  Forces  for  the  resultant  of  any  number  of  forces,  in 
lines  all  through  one  point. 

The  case  of  the  equilibrium  of  a  number  of  forces  aciing  at  one 
point,  is  evidently  deducible  at  once  from  this;  for  if  we  introduce 
one  other  force  equal  and  opposite  to  their  resultant,  this  v/iU  produce 
a  change  of  motion  equal  and  opposite  to  the  resultant  change  of 
motion  produced  by  the  given  forces ;  that  is  to  say,  will  produce  a 
condition  in  which  the  point  experiences  no  change  of  motion,  which, 
as  we  have  already  seeri,  is  the  only  kind  of  rest  of  v/hich  we  can  ever 
be  conscious. 

223.  Though  Newton  perceived  tiiat  the  Parallelogram  of  Forces, 
or  the  fundamental  principle  of  Statics,  is  essentially  involved  in  the 
second  law  of  motion,  and  gave  a  proof  which  is  virtually  the  'same  as 
the  preceding,  subsequent  writers  on  Statics  (especially  in  this  country) 
have  very  generally  ignored  the  fact ;  and  the  consequence  has  been 
the  introduction  of  various  unnecessary  Dynamical  Axioms,  more  or 
less  obvious,  but  in  reality  included  in  or  dependent  upon  Newton's 
laws  of  motion.  We  have  retained  Nev.ton's  method,  not  only  on 
account  of  its  admirable  simi)licity,  but  because  we  believe  it  contains 
the  most  philosophical  foundation  for  the  static  as  well  as  for  the 
kinetic  branch  of  the  dynamic  science. 

224.  But  the  second  law  gives  us  the  means  of  measuring  force, 
and  also  of  measuring  the  mass  of  a  body. 

For,  if  we  consider  the  actions  of  various  forces  upon  tiie  same 
body  for  equal  times,  we  evidently  have  changes  of  velocity  produced 
which  :v[Q proportional  to  the  forces.  The  changes  of  velocity,  then, 
give  us  in  this  case  the  means  of  comparing  the  magnitudes  of  different 
forces.  Thus  the  velocities  acquired  in  one  second  by  the  same  mass 
(falling  freely)  at  different  parts  of  the  earth's  surface,  give  us  the 
relative  amounts  of  the  earth's  attraction  at  these  places. 

Again,  if  equal  forces  be  exerted  on  different  bodies,  the  changes 
of  velocity  produced  in  equal  times  must  be  inversely  as  the  masses 
of  the  various  bodies.  This  is  approxiriiately  the  case,  for  instance, 
with  trains  of  various  lengths  started  by  the  same  locomotive  :  it  is 
exactly  realized  in  such  cases  as  the  action  of  an  electrified  body  on 
a  number  of  solid  or  hollow  spheres  of  the  same  .external  diameter, 
and  of  different  metals, 


DYNAMICAL  LAWS  AND  PRINCIPLES.  73 

Again,  if  we  find  a  case  in  which  different  bodies,  each  acted  on. 
by  a  force,  acquire  in  the  same  time  the  same  changes  of  velocity, 
the  forces  must  be  proportional  to  the  masses  of  the  bodies.  This, 
when  the  resistance  of  the  air  is  removed,  is  the  case  of  falling  bodies; 
and  from  it  we  conclude  that  the  weight  of  a  body  in  any  given 
locality,  or  the  force  with  which  the  earth  attracts  it,  is  proportional 
to  its  mass ;  a  most  important  physical  truth,  which  will  De  treated 
of  more  carefully  in  the  chapter  devoted  to  Properties  of  Matter. 

225.  It  appears,  lastly,  from  this  law,  that  every  theorem  of  Kine 
matics  connected  with  acceleration  has  its  counterpart  in  Kinetics. 
Thus,  for  instance  (§  38),  we  see  that  the  force  under  whicli  a  par- 
ticle describes  any  curve,  may  be  resolved  into  tv/o  components,  one 
in  the  tangent  to  the  curve,  the  other  toiuards  the  centre  of  curvature; 
their  magnitudes  being  the  acceleration  of  momentum,  and  the  pro- 
duct  of  the  momentum  and  the  angular  velocity  about  the  centre  of 
curvature,  respectively.  In  the  case  of  uniform  motion,  the  first  of 
these  vanisiies,  or  the  whole  force  is  perpendicular  to  the  direction 
of  motion.  When  there  is  no  force  perpendicular  to  the  direction 
of  motion,  there  is  no  curs'ature,  or  the  path  is  a  straight  line. 

226.  \Ve  have,  by  means  of  the  first  two  laws,  arrived  at  a  definition 
arid  a  measure  of  force;  and  have  also  found  how  to  compound,  and 
therefore  also  how  to  resolve,  forces  :  and  also  how  to  investigate 
the  motion  of  a  single  particle  subjected  to  given  forces.  But  more 
is  required  before  we  can  completely  understand  the  more  comple.x: 
cases  of  motion,  especially  those  in  which  we  have  mutual  actions 
between  or  amongst  two  or  more  bodies ;  such  as,  for  instance, 
attractions,  or  pressures,  or  transferrence  of  energy  in  any  form. 
This  is  perfectly  supplied  by 

227.  Lex  III.  Actioni  contrariarn  sanpcr  ci  acqualan  esse  rcadio- 
nem:  sive  corporum  duorum  actiones  in  se  mutuo  semper  esse  ae.quales 
ct  in  partes  contrarias  dirigi. 

To  ei'ery  action  there  is  ahvays  an  equal  and  contrary  reaction:  or,  tJ:c 
mutual  actio r,s  of  any  two  bodies  are  alxoays  equal  a/ul  oppositely  directed 

228.  If  one  body  presses  or  draws  aiiothcr,  it  is  pressed  or 
drawn  by  this  other  with  an  equal  force  in  the  opposite  direction. 
If  any  one  presses  a  stone  with  his  finger,  his  finger  is  pressed  with 
the  same  force  in  the  opposite  direction  by  the  stone.  A  horse 
towing  a  boat  on  a  canal  is  dragged  backwards  by  a  force  equal  to 
that  which  he  impresses  on  the  towing-rope  forwards.  By  whaiever 
amount,  and  in  whatever  direction,  one  body  has  its  motion  changed 
by  impact  upon  another,  this  other  body  has  its  motion  changed  by 
the  same  amount  in  the  opposite  direction;  for  at  each  ins'.ant  during 
the  impact  the  force  between  them  was  equal  and  opposite  on  the 
two.  When  neither  of  the  two  bodies  has  any  rotation,  whether 
before  or  after  impact,  the  changes  of  velocity  which  thuy  experience 
are  inversely  as  their  masses. 

When  one  body  attracts  another  from  a  distance,  this  other  atlracts 
it  with  an  equal  and  opposite  force.     This  law  holds  not  only  for 


74  PRELIMINAR  V. 

the  attraction  of  gravitation,  but  also,  as  Newton  himself  remarked 
and  verified  by  experiment,  for  magnetic  attractions  :  also  for  electric 
forces,  as  tested  by  Otto-Guericke. 

229.  What  precedes  is  founded  upon  Newton's  own  comments 
on  the  third  law,  and  the  actions  and  reactions  contemplated  are 
simple  forces.  In  the  scholium  appended,  he  makes  the  following 
remarkable  statement,  introducing  another  specification  of  actions 
and  reactions  subject  to  his  third  law,  the  full  meaning  of  which' 
seems  to  have  escaped  the  notice  of  commentators  : — 

Si  aSttimetur  agentis  actio  ex  ejus  vi  et  velociiate  conjunctim ;  et 
similiter  resistentis  readio  aestimefiir  conjunctim  ex  ejus  partium  singu- 
larum  velocitatibus  et  viribus  resistendi  ab  earitm  attritione^  cohaesione, 
J>ondere,  et  acceleratione  oriundis ;  erunt  actio  et  reactio,  *in  omni  instru- 
mentonmi  iisu,  sibi  invicem  semper  aequales. 

In  a  previous  discussion  Newton  has  shown  v/hat  is  to  be  under- 
stood by  the  velocity  of  a  force  or  resistance ;  i.  e.  that  it  is  the 
velocity  of  the  point  of  application  of  the  force  resolved  in  the  direction 
of  the  force,  in  fact  proportional  to  the  virtual  velocity.  Bearing  this- 
in  mind,  we  may  read  the  above  statement  as  follows  : — 

If  the  action  of  an  agent  be  measured  by  the  product  of  its  force  into 
its  velocity;  and  if,  similarly,  the  reactic?i  of  the  resistance  be  measured 
by  the  velocities  of  its  several  parts  into  their  several  forces,  whether 
these  arise  from  friction,  cohesion,  aveight,  or  acceleratio7i; — action  and 
reaction,  in  all  co?nbinations  of  machines,  will  be  equal  and  opposite. 

To  avoid  confusion  it  is  perhaps  better  to  use  the  word  Activity  as 
the  equivalent  o^  Actio  in  this  second  specification. 

Farther  on  we  shall  give  a  full  development  of  the  consequences- 
of  this  most  important  remark. 

230.  Newton,  in  the  passage  just  quoted,  points  out  that  forces 
of  resistance  against  acceleration  are  to  be  reckoned  as  reactions 
equal  and  opposite  to  the  actions  by  which  the  acceleration  is  pro- 
duced. Thus,  if  we  consider  any  one  material  point  of  a  system, 
its  reaction  against  acceleration  must  be  equal  and  opposite  to  the 
resultant  of  the  forces  which  that  point  experiences,  whether  by  the 
acncns  of  other  parts  of  the  sVstem  upon  it,  or  by  the  influence  of 
matter  not  belonging  to  the  system.  In  other  words,  it  must  be  in 
equilibrium  with  tliese  forces.  Hence  Newton's  view  am.ounts  to  this, 
that  all  the  forces  of  the  system,  with  the  reactions  against  accelera- 
tion of  the  material  points  com.posing  it,  form  groups  of  equiHbrating 
systems  for  these  points  considered  individually.  Hence,  by  the 
principle  of  superposition  of  forces  in  equilibrium,  all  the  forces 
acting  on  points  of  the  system  form,  with  the  reactions  against  acce- 
leration, an  equilibrating  set  of  forces  on  the  whole  system.  This 
is  the  celebrated  principle  first  explicitly  stated,  and  very  usefully 
applied,  by  D'Alembert  in  1742,  and  still  known  by  his  name.  We 
have  seen,  however,  that  it  is  very  distinctly  implied  in  Newton's 
own  interpretation  of  his  third  law  of  motion.    As  it  is  usual  to  inves- 


DYNAMICAL  LAWS  AND  PRINCIPLES,  75 

tigate  the  general  equations  or  conditions  of  equilibrium,  in  treatises 
on  Analytical  Dynamics,  before  entering  in  detail  on  the  kinetic 
branch  of  the  subject,  this  principle  is  found  practically  most  useful 
in  showing  how  we  may  write  down  at  once  the  equations  of  motion 
for  any  system  for  which  the  equations  of  equilibrium  have,  been 
investigated. 

231.  Every  rigid  body  may  be  imagined  to  be  divided  into  inde- 
finitely small  parts.  Now,  in  whatever  form  we  may  eventually 
find  a  physical  explanation  of  the  origin  of  the  forces  which  act 
between  these  parts,  it  is  certain  that  each  such  small  part  may  be 
considered  to  be  held  in  its  position  relatively  to  the  others  by  mutual 
forces  in  lines  joining  them. 

232..  From  this  we  have,  as  immediate  consequences  of  the  second 
and  third  laws,  and  of  the  preceding  theorems  relating  to  centre  of 
inertia  and  moment  of  momentum,  a  number  of  important  propo- 
sitions such  as  the  following  : — 

{a)  The  centre  of  inertia  of  a  rigid  body  moving  in  any  manner, 
but  free  from  external  forces,  moves  uniformly  in  a  straight  line. 

(J))  When  any  forces  whatever  act  on  the  body,  the  motion  of  the 
centre  of  inertia  is  the  same  as  it  would  have  been  had  these  forces 
been  applied  with  their  proper  magnitudes  and  directions  at  that 
point  itself. 

{c)  Since  the  moment  of  a  force  acting  on  a  particle  is  the  same 
as  the  moment  of  m.omentum  it  produces  in  unit  of  time,  the  changes 
of  moment  of  momentum  in  any  two  parts  of  a  rigid  body  due  to 
their  mutual  action  are  equal  and  opposite.  Hence  the  moment  of 
momentum  of  a  rigid  body,  about  any  axis  which  is  fixed  in  direction, 
and  passes  through  a  point  which  is  either  fixed  in  space  or  moves 
uniformly  in  a  straight  line,  is  unaltered  by  the  mutual  actions  of  the 
parts  of  the  hody. 

(d)  The  rate  of  increase  of  moment  of  momentum,  when  the  body 
is  acted  on  by  external  forces,  is  thie  sum  of  the  moments  of  these 
forces  about  the  axis. 

233.  We  shall  for  the  present  take  for  granted,  that  the  mutual 
action  between  two  rigid  bodies  may  in  every  case  be  imagined  as 
composed  of  pairs  of  equal  and  opposite  forces  in  straight  lines. 
From  this  it  follows  that  the  sum  of  the  quantities  of  motion,  parallel 
to  any  fixed  direction,  of  two  rigid  bodies  influencing  one  another 
in  any  possible  way,  remains  unchanged  by  their  mutual  action ; 
also  that  the  sum  of  the  moments  of  momentum  of  all  the  particles 
of  the  two  bodies,  round  any  line  iii  a  fixed  direction  in  space,  and 
passing  through  any  point  'moving  uniformly  in  a  straight  line  in  any 
direction,  remains  constant.  From  the  first  of  these  propositions  we 
infer  that  the  centre  of  inertia  of  any  number  of  mutually  influencing 
bodies,  if  in  motion,  continues  moving  uniformly  in  a  straight  line, 
unless  in  so  far  as  the  direction  or  velocity  of  its  motion  is  changed 
by  forces  acting  mutually  between  them  and  some  other  matter  not 
belonginK  to  them;  also  that  the  centre  of  inertia  of  any  body  or 


76  FRELIMJNARY, 

system  of  bodies  moves  just  as  all  their  matter,  if  concentrated  in 
a  point,  would  move  under  the  influence  offerees  equal  and  parallel 
to  the  forces  really  acting  on  its  different  parts.  From  the  second 
we  infer  that  the  axis  of  resultant  rotation  through  the  centre  of 
inertia  of  any  system  of  bodies,  or  through  any  point  either  at  rest 
or  moving  uniformly  in  a  straight  line,  remains  unchanged  in  direc- 
tion, and  the  sum  of  moments  of  momenta  round  it  remains  constant 
if  the.  system  experiences  no  force  from  without  This  principle 
is  sometimes  called  Conservation  of  Areas,  a  not  very  convenient 
designation.  From  this  principle  it  follows  that  if  by  internal  action 
such  as  geological  upheavals  or  subsidences,  or  pressure  of  the  winds 
on  the  water,  or  by  evaporation  and  rain-  or  snow-fall,  or  by  any  in- 
fluence not  depending  on  the  attraction  of  sun  or  moon  (even  though 
dependent  on  solar  heat),  the  disposition  of  land  and  water  becomes 
altered,  the  component  round  any  fixed  axis  of  the  moment  of  mo- 
mentum, of  the  earth's  rotation  remains  constant. 

234.  The  kinetic  energy  of  any  system  is  equal  to  the  sum  of  the 
kinetic  energies  of  a  mass  equal  to  the  sum  of  the  masses  of  the 
system,  moving  with  a  velocity  equal  to  that  of  its  centre  of  inertia, 
and  of-  the  motions  of  the  separate  parts  relatively  to  the  centre,  of 
inertia. 

Let  (?/ represent  the  velocity  of  the  centre  of  inertia,  IP  that  of 

3  any  point  of  the  system  relative  to  0.     Then 

^J'  the  actual  velocity  of  that  point  is  OP^  and  the 

_^,^''W''^i  proof  of  §  196  applies  at  once — it  being  re- 

■^      /    \  membered  that  the  mean  of  /(2,  i-  e.  the  mean 

.,*£:f!ll_ — /.. i.  of  the  velocities  relative  to  the  centre  of  inertia 

^              i          Q,  ^^.^^  parallel  to  01,  is  zero  by  §  65. 

235.  The  kinetic  energy  of  rotation  of  a  rigid  system  about  any; 
axis  is  {§§55,  179)  expressed  by  |2wrV,  where  m  is  the  mass  of 
any  part,  r  its  distance  from  the  axis,  and  w  the  angular  velocity  of 
rotation.  It  may  evidently  be  written  in  the  form  \u?1m7^.  The 
factor  2?«r*  is  of  course  (§  198)  the  Moment  of  Inertia  of  the  system 
about  the  axis  in  question. 

It  is  worth  while  to  notice  that  the  moment  of  momentum  of  any 
rigid  system  about  an  axis,  being  "^rnvr  -ai^mr^,  is  the  product  of 
the  angular  velocity  into  the  moment  of  inertia;  while,  as  above,  the 
half  product  of  the  moment  of  inertia  by  the  square  of  the  angular 
velocity  is  the  kinetic  energy. 

If  we  take  a  quantity  k,  such  that 

k'%m  -  ^vir", 

k  is  called  the  Radius  of  Gyration  about  the  axis  from  which  r  is 
measured.  The  radius  of  gyration  about  any  axis  is  therefore  the 
distance  from  that  axis  at  which,  if  the  whole  mass  were  placed,  it 
would  have  the  same  moment  of  inertia  as  before.  In  a  fly-wheel, 
where  it  is  desirable  to  have  as  great  a  moment  of  inertia  with  as 
small  a  mass  as  possible,  within  certain  limits  ol  dimensions,  the 


DYNAMICAL  LAWS  AND  PRINCIPLES.  77 

greater  part  of  the  mass  is  formed  into  a  ring  'of  the  largest  admis* 
sible  diameter,  and  the  radius  of  this. ring  is  then  approximately  the 
radius  of  gj-ration  of  the  whole. 

236.  The  rate  of  increase  of  moment  of  m.omentum  is  thus,  in  New- 
ton's notation  (§  2S),  wSwr*;  and,  in  the  case  of  a  body  free  to  rotate 
about  a  fixed  axis,  is  equal  to  the  moment  of  the  couple  about  that 
axis;     Hence  a  constant  couple  gives  uniform  acceleration  of  angular 

velocity;  or  (1)  =  —™-.     By  §  178  we  see  that  the  corresponding 

,       ....     Force 
formula  for  hnear  acceleration  \%s  =  v=-  -■.-- 

237.  For  every  rigid  body  there  may  be  described  about  any  point 
as  centre,  an  ellipsoid  (called  Foinsofs  Momental  Ellipsoid)  which  is 
such  that  the  length  of  any  radius-vector  is  inversely  proportional  to 
the  radius  of  gyration  of  the  body  about  that  radius-v6ctor  as  axis. 

TI)e  axes  of  the  ellipsoid  are  ih.Q.  Principal  Axes  hi  inertia  of  the 
body  at  the  point  in  question. 

When  the  moments  of  inertia  about  two  of  these  are  equal,  the 
eUipsoid  becomes  a  spheroid,  and  the  radius  of  gyration  is  the  same 
for  ever)'  axis  in  the  plane  of  its  equator. 

When  all  three  principal  moments  are  equal,  the  ellipsoid  becomes 
a  sphere,  and  every  axis  has  the  same  radius  of  gyration. 

238.  The  principal  axes  at  any  point  of  a  rigid  body  are  normals 
to  the  three  surfaces  of  the  second  order  which  pass  through  that 
point,  and  are  confocal  with  an  ellipsoid,  having  its  centre  at  the 
centre  of  inertia,  and  its  three  principal  diameters  coincident  with  the 
three  principal  axes  through  these  points,  and  equal  respectively  to 
the  doubles  of  the  radii  of  gyration  round  them.  This  ellipsoid  is 
called  the  Central  Ellipsoid. 

239.  A  rigid  body  is  said  to  be  kinetically  sjTnmetrical  about  its 
centre  of  inertia  when  its  moments  of  inertia  about  three  principal 
axes  through  that  point  are  equal ;  and  therefore  necessarily  the 
moments  cf  inertia  about  all  axes  through  that  point  equal  .(§  237), 
and  all  these  axes  principal  axes.  About  it  uniform  spheres,  cubes, 
and  in  general  any  complete  crystalline  solid  of  the  first  system  (see 
chapter  on  Properties  of  Matter)  are  kinetically  symmetrical. 

A  rigid  body  is  kinetically  5)'mmetrical  about  an  axis  when  this 
axis  is  one  of  the  principal  axes  through  tlie  centre  of  inertia,  and 
the  moments  of  inertia  about  the  other  two,  and  therefore  about  any 
line  in  their  plane,  are  equal.  A  spheroid,  a  square  or  equilateral 
triangular  prism  or  plate,  a  circular  ring,  disc,  or  cylinder,  or  any 
complete  cr)-stal  of  the  second  or  fourth  system,  is  kinetically  sm- 
metrical  about  its  axis. 

240.  The  foundation  of  the  abstract  theory  of  energ>'  is  laid  by 
Newton  in  an  admirably  distinct  and  compact  manner  in  the  sentence 
of  his  scholium  already  quoted  (§  aap),  in  which  he  points  out  its 


78  PRELIMINARY. 

application  to  mechanics'.  The  actio  agentis,  as  he  defines  it,  which 
IS  evidently  equivalent  to  the  product  of  the  effective  component  of 
the  force,  into  the  velocity  of  the  pomt  on  which  it  acts,  is  simply,  in 
modern  English  phraseology,  the  rate  at  which  the  agent  works.  The 
subject  for  measurement  here  is  precisely  the  same  as  that  for  which 
•Watt,  a  hundred  years  later,  introduced  the  practical  unit  of  a  '^ Horse- 
power,' or  the  rate  at  which  an  agent  works  when  overcoming  33,000 
times  the  weight  of  a  pound  through  the  space  of  a  foot  m  a  minute; 
that  IS,  producing  550  foot-pounds  of  work  per  second.  The  unit, 
however,  which  is  most  generally  convenient  is  that  which  Newton's 
definition  implies,  namely,  the  rate  of  doing  work  in  which  the  unit 
of  energy  is  produced  in  the  unit  of  time. 

241.  Looking  at  Newton's  words  (§  229)  in  this  light,  we  see  that 
they  may  be  logically  converted  into  the  following  form  : — 

Work  done  on  any  system  of  bodies  (in  Newton's  statement,  the  parts 
of  any  machine)  has  its  equivalent  in  work  done-  against  friction, 
molecular  forces,  or  gravity,  if  there  be  no  acceleration ;  but  if  there 
be  acceleration,  part  of  the  work  is  expended  in  overcoming  the  resistance 
to  acceleration,  and  the  additional  kinetic  energy  developed  is  equivalent 
to  the  work  so  spent.     This  is  evident  from  §  180. 

When  part  of  the  work  is  done  against  molecular  forces,  as  in 
bending  a  spring;  or  against  gravity,  as  in  raising  a  weight;  the 
recoil  of  the  spring,  and  the  fall  of  the  weight,  are  capable  at  any 
future  time,  of  reproducing  the  work  originally  expended  (§  207), 
But  in  Newton's  day,  and  long  afterwards,  it  was  supposed  that  work 
was  absolutely  lost  by  friction,  and,  indeed,  this  statement  is  still  to 
be  found  even  in  recent  authoritative  treatises.  But  we  must  defer 
the  examination  of  this  point  till  we  consider  in  its  modern  form  the 
principle  of  Conservation  of  Energy. 

242.  If  a  system  of  bodies,  given  either  at  rest  or  in  motion,  be 
influenced  by  no  forces  from  without,  the  sum  of  the  kinetic  energies 
of  all  its  parts  is  augmented  in  any  time  by  an  amount  equal  to  the 
whole  work  done  in  that  time  by  the  mutual  forces,  which  we  may 
imagine  as  acting  between  its  points.  When  the  lines  in  which  these 
forces  act  remain  all  unchanged  in  length,  the  forces  do  no  work,  and 
the  sum  of  the  kinetic  energies  of  the  whole  system  remains  constant. 
If,  on  the  other  hand,  one  of  these  lines  varies  in  length  during  the 
motion,  the  mutual  forces  in  it  will  do  work,  or  will  consume  work, 
according  as  the  distance  varies  with  or  against  them. 

243.  A  limited  system  of  bodies  is  said  to  be  dynamically  con- 
.servative  (or  simply  conservative,  when  force  is  understood  to  be  the 
•subject),  if  the  mutual  forces  between  its  parts  always  perform,  or 
.always  consume,   the  same   amount   of  work   during  any   motion 

*  The  reader  will  remember  that  we  use  the  word  'mechanics'  in  its  true  classical 
sense,  the  science  of  machines,  the  sense  in  which  Newton  himself  used  it,  when  he 
•dismissed  the  further  consideration  of  it  by  saying  (in  the  scholium  referred  to), 
•CaeUrum  mtchanicam  trnctare  non  est  hujus  insktuti.  ' 


DYNAMICAL  LAWS 'AND  PRINCIPLES.  79 

whatever,  by  which  it  can  pass  fiom  one  particular  configuration 
to  another. 

244.  The  whole  theory  of  energy  in  physical  science  is  founded 
on  the  following  proposition  : — 

If  the  mutual  forces  between  the  parts  of  a  material  system  are 
independent  of  their  velocities,  whether  relative  to  one  another,  or 
relative  to  any  external  matter,  the  system  must  be  dynamically 
conservative. 

For  if  more  work  is  done  by  the  mutual  forces  on  the  difierent 
parts  of  the  system  in  passing  from  one  particular  configuration  to 
another,  by  one  set  of  paths  than  by  another  set  of  paths,  let  the 
system  be  directed,  by  frictionless  constraint,  to  pass  from  the  first 
configuration  to  the  second  by  one  set  of  paths  and  return  by  the 
other,  over  and  over  again  for  ever.  It  will  be  a  continual  source  of 
energy  without  any  consumption  of  materials,  which  is  impossible. 

245.  The  potential  energy  of  a  conservative  system,  in  the  confi- 
guration which  it  has  at  any  instant,  is  the  amount  of  work  that  its 
mutual  forces  perform  during  the  passage  of  the  system  from  any 
one  chosen  configuration  to  the  configuration  at  the  time  referred  to. 
It  is  generally,  but  not  always,  convenient  to  fi.x  the  particular  con- 
figuration chosen  for  the  zero  of  reckoning  of  potential  energy,  so 
that  the  potential  energ)',  in  every  other  configuration  practically 
considered,  shall  be  positive. 

248.  The  potential  energ>'  of  a  conservative  system,  at  any  instant, 
depends  solely  on  its  configuration  at  that  instant,  being,  according  to 
definition,  the  same  at  all  times  when  the  system  is  brought  again 
and  again  to  the  same  configuration.  It  is  therefore,  m  mathematical 
language,  said  to  be  a  function  of  the  co-ordinates  l)y  v.-hich  the 
positions  of  the  different  parts  of  the  system  are  specified.  If",  for 
example,  we  have  a  conservative  system  consisting  of  two  material 
points  ;  or  tv/o  rigid  bodies,  acting  upon  one  another  v^'ith  force 
dependent  only  on  the  relative  position  of  a  point  belonging  to  one 
of  them,  and  a  pomt  belonging  to  the  other ;  the  potential  energy 
of  the  system  depends  upon  the  co-ordinates  of  one  of  these  points 
relatively  to  lines  of  reference  in  fixed  directions  through  the  other. 
It  will  therefore,  in  general,  depend  on  three  independent  co-ordi- 
nates, which  we  may  conveniently  take  as  the  distance  between  the 
two  points,  and  two  angles  specifying  the  absolute  direction  of  the 
line  joining  them.  Thus,  for  example,  let  the  bodies  be  two  uniform 
metal  globes,  electrified  with  any  given  quantities  of  electricity,  and 
placed  lu  an  insulating  medium  such  as  air,  in  a  region  of  space 
under  the  influence  of  a  vast  distant  electrified  body.  The  mutual 
action  between  these  two  spheres  will  depend  solely  on  the  relative 
position  of  their  centres.  It  will  consist  partly  of  gravitation,  de- 
pending solely  on  the  distance  between  their  centres,  and  of  electric 
force,  which  Will  depend  on  the  distance  between  them,  but  also,  ia 
virtue  of  the  inductive  action  of  the  distant  body,  will  depend  on  the 
absolute  direction  of  the  line  joining  their  centres.     Or  again,  if  the 


8o  PRELIMINARY. 

system  consist  of  two  balls  of  soft  iron,  in  any  locality  of  the  earth's 
surface,  their  mutual  action  will  be  partly  gravitation,  and  partly 
due  to  the  magnetism  induced  in  them  by  terrestrial  magnetic  force. 
The  portion  of  the  potential  energy  depending  on  the  latter  cause, 
will  be  a  function  of  the  distance  between  their  centres  and  the  in- 
clination of  this  line  to  the  direction  of  the  terrestrial  magnetic  force. 

247.  In  nature  the  hypothetical  condition  of  §  243  is  apparently 
violated  \n  all  circumstances  of  motion.  A  material  system  can  never 
be  brought  through  any  returning  cycle  of  motion  without  spending 
more  work  against  the  mutual  forces  of  its  parts  than  is  gained  from 
these  forces,  because  no  relative  motion  can  take  place  without 
meeting  with  frictional  or  other  forms  of  resistance ;  among  which 
are  included  (i)  mutual  friction  between  solids  sliding  upon  one 
another;  (2)  resistances  due  to  the  viscosity  of  fluids,  or  imperfect 
elasticity  of  solids;  (3)  resistances  due  to  the  induction  oi  electric 
currents ;  (4)  resistances  due  to  varying  magnetization  under  the 
influence  of  imperfect  magnetic  retentiveness.  No  motion  in  nature 
can  take  place  v^ithout  meeting  resistance  due  to  some,  if  not  to  all, 
of  these  influences.  It  isrnatter  of  everyday  experience  that  friction 
and  imperfect  elasticity  of  solids  impede  the  action  of  all  artificial 
mechanisms,  and  that  even  when  bodies  are  detached,  and  left  to 
move  freely  in  the  air,  as  falling  bodies,  or  as  projectiles,  they  expe 
rience  resistance  owing  to  the  viscosity  of  the  air. 

The  greater  masses,  planets  and  comets,  moving  in  a  less  resisting 
medium,  show  less  indications  of  resistance'.  Indeed  it  cannot  be  said 
that  observation  upon  any  one  of  these  bodies,  with  the  possible  excep- 
tion of  tlncke's  comet,  has  demonstrated  resistance.  But  the  analogies 
of  nature,  and  the  ascertained  facts  of  physical  science,  forbid  us  to 
doubt  that  every  one  of  them,  every  star,  and  every  body  of  any  kind 
moving  in  any  part  of  space,  has  its  relative  motion  impeded  by  the 
-air,  gas,  vapour,  medium,  or  whatever  we  choose  to  call  the  substance 
occupying  the  space  immediately  round  it;  just  as  the  motion  of  a 
rifle-bullet  is  impeded  by  the  resistance  of  the  air. 

248.  There  are  also  indirect  resistances,  owing  to  friction  impeding 
the  tidal  motions,  on  all  bodies  which,  like  the  earth,  have  portions 
of  their  free  surfaces  covered  by  liquid,  which,  as  long  as  these  bodies 
move  relatively  to  neighbouring  bodies,  must  keep  drawing  off  energy 
from  their  relative  motions.  Thus,  if  we  consider,  in  the  first  place, 
the  action  of  the  moon  alone,  on  the  earth  with  its  oceans,  lakes,  and 
rivers,  we  perceive  that  it  must  tend  to  equalize  the  periods  of  the 
earth's  rotation  about  its  axis,  and  of  the  revolution  of  the  two  bodies 
about  their  centre  of  inertia  ;  because  as  long  as  these  periods  differ, 
the  tidal  action  of  the  earth's  surface  must  keep  subtracting  energy 
from  their  motions.  To  view  the  subject  more  in  detail,  and,  at  the 
same  time,  to  avoid  unnecessary  complications,  let  us  suppose  the 

*  Newton,  Principia.  (Remarks  on  the  first  law  of  motion.)  'Majora  autem 
Planetarum  et  Cometarum  corpora  motus  suos  et  progressivoa  et  circulares,  in 
spatiis  minus  resistentibus  factos,  conservar.t  diutius.' 


DYNAMICAL  LA  WS  AND  PRINCIPLES.  8r 

moon  to  be  a  uniform  sphencal  body.  The  mutual  action  and 
reaction  of  gravitation  between  her  mass  and  the  earth's,  will  be 
equivalent  to  a  single  force  in  some  line  through  her  centre ,  and 
must  be  such  as  to  impede  the  earth's  rotation  as  long  as  this  is 
performed  in  a  shorter  period  than  the  moon's  motion  round  the 
earth.  It  must  therefore  lie  in  some  such  direction  as  the  line  M(^ 
in  the  diagram,  which  represents,  necessarily 
wnth  enormous  exaggeration,  its  deviation, 
OQ,  from  the  earth's  centre.  Now  the  actual 
force  on  the  moon  in  the  line  MQ.  may  be 
regarded  as  consisting  of  a  force  in  the  line 
MO  towards  the  earth's  centre,  sensibly 
equal  in  amount  to  the  whole  force,  and  a 
comparatively  very  small  force  m  the  line 
MT  perpendicular  to  MO  This  latter  is 
very  nearly  tangential  to  the  moon's  path, 
and  is  m  the  direction  Tinlh  her  motion. 
Such  a  force,  if  suddenly  commencing  to  act,  would,  in  the  first  place, 
increase  the  moon's  velocity;  but  after  a  certain  time  she  would  have 
moved  so  much  farther  from  the  earth,  in  virtue  of  this  acceleration, 
as  to  have  lost,  by  moving  against  the  earth's  attraction,  as  much 
velocity  as  she  had  gained  by  the  tangential  accelerating  force.  The 
integral  effect  on  the  moon's  motion,  of  the  particular  disturbing 
cause  now  under  consideration,  is  most  easily  found  by  using  the  prin- 
ciple of  moments  of  momenta  (S  233).  Thus  we  see  that  as  much 
moment  of  momentum  is  gained  in  any  time  by  the  motions  of  the 
centres  of  inertia  of  the  moon  and  earth  relatively  to  their  common 
centre  of  inertia,  as  is  lost  by  the  earth's  rotation  about  its  axis.  It 
is  found  that  the  distance  would  be  iriCreascd  to  about  347,100  miles, 
and  the  period  lengthened  to  4S'36  days.  Were  there  no  other  body 
in  the  universe  but  the  earth  and  the  moon,  tiiese  two  bodies  might 
go  on  ir.oving  thus  for  ever,  in  circular  orbits  round  their  common 
centre  of  inertia,  and  the  earth  rotating  about  its  axis  in  the  same 
period,  so  as  always  to  turn  the  same  face  to  the  moon,  and  therefore 
to  have  all  the  li';uids  at  its  surface  at  rest  relatively  to  the  solid.  But 
the  existence  of  the  sun  would  prevent  any  such  state  of  things  from 
being  permanent.  There  \/ouId  be  solar  tides — twice  high  v.-ater  and 
twice  lov.'  water — in  the  period  of  the  earth's  revolution  relatively  to 
the  sun  (that  is  to  say,  twice  in  the  solar  day,  or,  which  would  be  the 
same  thing,  the  month).  This  could  not  go  on  without  loss  of  energy 
b\-  fluid  friction.  It  is  not  easy  to  trace  the  whole  course  of  ihe 
disturbance  in  the  earth's  and  moon's  motions  which  this  cause 
would  produce,  but  its  ultimate  eflect  must  be  to  bnng  the  earth, 
moon,  and  sun  to  rotate  round  their  common  centre  of  inertia,  like 
parts  of  one  rigid  body.  It  is  probable  that  the  moon,  in  ancient 
times  hquid  or  viscous  in  its  outer  layer  if  not  throughout,  was  thus 
brought  to  turn  alwa)s  the  same  face  to  the  earth. 

249.    'We  have  no  data  in  the  present  state  of  science  for  estimating 
the  relative  importance  of  tidal  friction,  and  of  the  resistance  of  die 


8a  PRELIMINARY. 

resisting  medium  through  which  the  earth  and  moon  move;  but  what- 
■ever  it  may  be,  there  can  be  but  one  ultimate  result  for  such  a  system 
as  that  of  the  sun  and  planets,  if  continuing  long  enough  under  ex- 
isting laws,  and  not  disturbed  by  meeting  with  other  moving  masses 
in  space.  That  result  is  the  falling  together  of  all  into  one  mass, 
which,  although  rotating  for  a  time,  must  in  the  end  come  to  rest 
relatively  to  the  sunounding  medium. 

250.  The  theory  of  energy  cannot  be  completed  until  we  are  able 
to  examine  the  physical  influences  which  accompany  loss  of  energy 
in  each  of  the  classes  of  resistance  mentioned  above  (§  247).  We 
shall  then  see  that  in-.every  case  in  which  energy  is  lost  by  resistance, 
heat  is  generated;  and  we  shall  learn  from  Joule's  investigations  that 
the  quantity  of  heat  so  generated  is  a  perfectly  definite  equivalent  for 
the  energy  lost.  Also  that  in  no  natural  action  is  there  ever  a  develoi> 
ment  of  energy  which  cannot  be  accounted  for  by  the  disappearance 
of  an  equal  amount  elsewhere  by  means  of  some  known  physical 
agency.  Thus  we  shall  conclude,  that  if  any  limited  portion  of  the 
material  universe  could  be  perfectly  isolated,  so  as  to  be  prevented 
from  either  giving  energy  to^  or  taking  energy  from,  matter  external 
to  It,  the  sum  of  its  potential  and  kinetic  energies  would  be  the  same 
at  all  times:  m  other  words,  that  every  material  system  subject  to  no 
other  forces  than  actions  and  reactions  between  its  parts,  is  a  dyna- 
mically conservative  system,  as  defined  above  (§  243).  But  it  is  only 
when  the  inscrutably  mmute  motions  among  small  parts,  possibly  the 
ultimate  molecules  of  matter,  which  constitute  light,  heat,  and  mag- 
netism; and  the  intermolecular  forces  of  chemical  affinity,  are  taken 
into  account,  along  v/ith  the  palpable  motions  and  measurable  forces 
of  which  we  become  cognizant  by  direct  observation,  that  we  can 
recognize  the  universally  conservative  character  of  all  natural  dynamic 
action,  and  perceive  the  bearing  of  the  principle  of  reversibility  on  the 
whole  class  of  natural  actions  involving  resistance,  which  seem  to 
violate  it.  In  the  meantime,  in  our  studies  of  abstract  dynamics,  it 
will  be  sufficient  to  introduce  a  special  reckoning  for  energy  lost  in 
working  against,  or  gamed  from  work  done  by,  forces  not  belonging 
palpably  to  the  conservative  class. 

251.  The  only  actions  and  reactions  between  the  parts  of  a  system, 
not  belonging  palpably  to  the  conservative  class,  which  we  shall  con- 
sider in  abstract  dynamics,  are  those  of  fnction  between  solids  sliding 
on  solids,  except  m  a  few  instances  in  which  wc  shall  consider  the 
general  character  and  ultimate  results  of  effects  produced  by  viscosity 
of  fluids,  imperfect  elasticity  of  solids,  imperfect  electric  conduction, 
or  imperfect  magnetic  retentivepess.  We  shall  also,  m  abstract  dyna- 
mics, consider  forces  as  applied  to  parts  of  a  limited  system  arbitrarily 
from  without.     These  vve  shall  call,  for  brevity,  the  apj)lied  forces. 

252.  The  law  of  energy  may  then,  in  abstract  dynamics,  be  ex- 
pressed as  follows  • — 

The  whole  work  done  in  any  time,  on  any  limited  material  system, 
by  apphed  forces,  is  equal  to  the  vvhole  efi'ect  in  the  forms  of  potential 


DYNAMICAL  LAWS  AND  PRINCIPLES.  83 

and  kinetic  energy  produced  in  the  system,  together  with  the  work  lost 
in  friction. 

253.  This  prin'ciple  may  be  regarded  as  comprehending  the  whole 
of  abstract  dynamics,  because,  as  we  now  proceed  to  show,  the  con- 
ditions of  equilibrium  and  of  motion,  in  every  possible  case,  may  be 
derived  from  iL 

254.  A  material  system,  whose  relative  motions  are  unresisted  by 
friction,  is  in  equilibrium  in  any  particular  configuration  if,  and  is  not 
in  equilibrium^  unless,  the  rate  at  which  the  applied  forces  perform 
work  at  the  instant  of  passing  through  it  is  equal  to  that  at  which 
potential  energy  is  gained,  in  every  possible  motion  through  that 
configuration.  This  is  the  celebrated  principle  of  virtual  velocities 
which  Lagrange  made  the  basis  of  his  Mkanique  Analytiquc. 

255.  To  prove  it,  we  have  first  to  remark  that  the  system  cannot 
possibly  move  away  from  any  particular  configuration  except  by  work 
being  done  upon  it  by  the  forces  to  which  it  is  subject:  it  is  therefore 
in  equilibrium  if  the  stated  condition  is  fulfilled.  To  ascertain  tliat 
nothing  less  than  this  condition  can  secure  the  equilibrium,  let  us 
first  consider  a  system  having  only  one  degree  of  freedom  to  move. 
Whatever  forces  act  on  the  whole  system,  we  may  always  hold  it  in 
equilibrium  by  a  single  force  applied  to  any  one  point  of  the  system 
in  its  line  of  motion,  opposite  to  the  direction  in  which  it  tends  to 
move,  and  of  such  magnitude  that,  in  any  infinitely  small  rtoticn  in 
either  direction,  it  shall  resist,  or  shall  do,  as  much  work  as  the  other 
forces,  whether  applied  or  internal,  altogether  do  or  resist.  Now,  by 
the  principle  of  superposition  of  forces  in  equilibrium,  we  might, 
without  altering  their  effect,  apply  to  any  one  point  of  the  system  such 
a  force  as  we  have  just  seen  would  hold  the  system  in  equilibrium,  and 
another  force  equal  and  opposite  to  it.  All  the  other  forces  being 
balanced  by  one  of  these  two,  they  and  it  might  again,  by  the  principle 
of  superposition  of  forces  in  equilibrium,  be  removed;  and  therefore 
the  whole  set  of  given  forces  would  produce  the  same  effect,  whether 
for  equilibrium  or  for  motion,  as  the  single  force  which  is  left  acting 
alone.  This  single  force,  since  it  is  in  a  line  in  which  the  point  of  its 
application  is  free  to  move,  must  move  the  system.  Hence  the  given 
forces,  to  which  the  single  force  has  been  proved,  equivalent,  cannot 
possibly  be  in  equilibrium  unless  their  whole  work  for  an  infinitely 
small  motion  is  nothing,  in  which  case  the  single  equivalent  force  is 
reduced  to  nothing.  But  whatever  amount  of  freedom  to  move  the 
whole  system  may  have,  we  may  always,  by  the  application  of  fnc- 
tionless  constraint,  limit  it  to  one  degree  of  freedom  only; — and  this 
may  be  freedom  to  execute  any  particular  motion  whatever,  possible 
under  the  given  conditions  of  the  system.  If,  therefore,  in  any  such 
infinitely  small  motion,  there  is  variation  of  potential  energ}'  uncom- 
pensated by  vvork  of  the  applied  forces,  constraint  limiting  th^  freedom 
of  the  system  to  only  this  motion  will  bnng  us  to  the  case  m  which  we 
have  just  demonstrated  there  cannot  be  equilibrium.    But  the  applica- 


84  PRELIMINARY. 

tion  of  constraints  limiting  motion  cannot  possibly  disturb  equilibrium, 
and  therefore  the  given  system  under  the  actual  conditions  cannot  be 
in  equilibrium  in  any  particular  configuration  if  the  rate  of  doing  work 
is  greater  than  that  at  which  potential  energy  is  stored  up  in  any  pos- 
sible motion  through  that  configuration. 

256.  If  a  material  system,  under  the  influence  of  mternal  and 
appHed  forces,  varying  according  to  som.e  definite  law,  is  balanced 
by  them  in  any  position  in  which  it  may  be  placed,  its  equilibrium  is 
said  to  be  neutral.  This  is  the  case  with  any  spherical  body  of 
uniform  material  resting  on  a  horizontal  plane.  A  right  cylinder  or 
cone,  bounded  by  plane  ends  perpendicular  to  the  axis,  is  also  in 
neutral  equilibnum  on  a  horizontal  plane.  Practically,  any  mass  of 
moderate  dimensions  is  in  neutral  equilibrium  when  its  centre  of 
inertia  only  is  foced,  since,  when  its  longest  dimension  is  small  in 
comparison  with  the  earth's  radius,  gravity  is,  as  we  shall  see,  ap- 
proximately equivalent  to  a  single  force  through  this  pomt. 

But  if,  when  displaced  infinitely  little  m  any  direction  from  a  par- 
ticular position  of  equilibrium,  and  left  to  itself,  it  commences  and 
■  continues  vibrating,  without  ever  experiencing  more  than  infinitely 
small  deviation  in  any  of  its  parts,  from  the  position  of  equilibrium, 
the  equilibnum  m  this  position  is  said  to  be  stable.  A  weight  sus- 
pended by  a  string,  a  uniform  sphere  m  a  hollow  bowl,  a  loaded  sphere 
resting  on  a  horizontal  plane  with  the  loaded  side  lowest,  an  oblate 
body  resting  with  one  end  of  its  shortest  diameter  on  a  honzontai 
plane,  a  plank,  whose  thickness  is  small  compared  v^^ith  its  length  and 
breadth,  floating  on  water,  are  all  cases  of  stable  equilibrium;  if  we 
neglect  the  motions  of  rotation  about  a  vertical  axis  m  the  second, 
third,  and  fourth  cases,  and  honzontai  motion  in  general,  in  the  fifth, 
for  all  of  which  the  equilibrium  is  neutral. 

If,  on  the  other  band,  the  system  can  be  displaced  in  any  way  from 
a  position  of  equilibrium,  so  that  when  left  to  itself  it  will  not  vibrate 
within  infinitely  small  limits  about  the  position  of  equilibrium,  but  will 
move  farther  and  farther  away  from  it,  the  equilibrium  m  this  position 
is  said  to  be  unstable.  Thus  a  loaded  sphere  resting  on  a  horizontal 
plane  v/ith  its  load  as  high  as  possible,  an  egg-s?iaped  body  standing 
on  one  end,  a  board  floating  edgewise  in  water,  would  present,  if  they 
could  be  realized  in  practice,  cases  of  unstable  equilibrium^. 

When,  as  in  many  cases,  the  nature  of  the  equilibrium  varies  with 
the  direction  of  displacement,  if  unstable  for  any  possible  displace- 
ment it  is  practically  unstable  on  the  whole.  Thus  a  circular  disc 
standing  on  its  edge,  though  in  neutral  equilibrium  for  displacements 
in  its  plane,  yet  being  in  unstable  equilibrium  for  those  perpendicular 
to  its  plane,  is  practically  unstable.  A  sphere  resting  in  equilibnum  on 
a  saddle  presents  a  case  in  which  there  is  stable,  neutral,  or  unstable 
equilibnum,  according  to  the  direction  in  which  it  may  be  displaced 
by  rolling;  but  practically  it  is  unstable. 

257.  The  theorj-  of  energy  shows  a  very  clear  and  simple  test  for 
discriminating  these  characters,  or  determining  whether  the  equilibnum 


DYNAMICAL  LAWS  AND  PRINCIPLES,  85 

is  neutral,  stable,  or  unstable,  in  any  case.  If  there  is  just  as  much 
potential  energy  stored  up  as  there  is  work  performed  by  the  applied  and 
internal  forces  in  any  possible  displacement,  the  equilibrium  is  neutral, 
but  not  unless.  If  in  every  possible  infinitely  small  displacement 
from  a  position  of  equilibnura  there  is  more  potential  energy  stored 
up  than  work  done,  the  equilibrium  is  thoroughly  stable,  and  not 
unless.  If  m  any  or  m  every  mfinitely  small  displacem.ent  ffom  a 
position  of  equilibrium  there  is  more  work  done  than  energy  stored 
up,  the  equilibrium  is  unstable.  It  follows  that  if  the  system  is  in- 
fluenced only  by  internal  forces,  or  if  the  applied  forces  follow  the 
law  of  doing  always  the  same  amount  of  work  upon  the  system  pass- 
ing from  one  configuration  to  another  by  all  possible  paths,,  the  whole 
potential  energy  must  be  constant,  in  all  positions,  for  neutral  equili- 
brium; must  be  a  minimum  for  positions  of  thoroughly  stable  equili- 
brium ;  must  be  either  a  maximum  for  all  displacements,  or  a  maximum 
for  some  displacements  and  a  minimum  for  others,  when  there  is 
unstable  equilibrium 

258.  We  have  seen  that,  according  to  D'Alembert's  principle,  as 
explained  above  (§  230),  forces  acting  on  the  different  points  of  a 
material  system,  and  their  reactions  against  the  accelerations  which 
they  actually  experience  in  any  case  of  motion,  are  in  equilibrium 
with  one  another.  Hence  in  any  actual  cnse  of  motion,  not  only  is 
the  actual  work  done  by  the  forces  equal  to  the  kinetic  energy  pro- 
duced in  any  infinitely  small  time,  m  virtue  of  the  actual  accelerations , 
but  so  also  is  the  work  which  would  be  done  by  the  forces,  in  any 
infinitely  small  tmie,  if  the  velocities  of  the  points  constituting  the 
system  were  at  any  instant  changed  to  any  possible  infinitely  small 
velocities,  and  the  accelerations  unchanged.  This  statement,  when 
put  into  the  concise  language  of  mathematical  analysis,  constitutes 
Lagrange's  application  of  the  'principle  of  virtual  velocities'  to  ex- 
press the  conditions  of  D'Alembert's  equuibnum  betv.-een  the  forces 
acting,  and  the  resistances  of  the  masses  to  acceleration.  It  com- 
jjrehends,  as  we  have  seen,  every  possible  condition  of  every  case  of 
motion.  The  'equations  of  motion'  in  any  particular  case  are,  as 
Lagrange  has  shown,  deduced  from  it  with  great  ease 

259.  When  two  bodies,  m  relative  motion,  come  mto  contact, 
pressure  begins  to  act  between  them  to  prevent  any  parts  of  them 
from  jointly  occupying  the  same  space.  This  force  commences  from 
nothing  at  the  first  point  of  collision,  and  gradually  increases  per  unit 
of  area  on  a  gradually  increasing  surface  of  contact.  If,  as  is  always 
the  case  in  nature,  each  body  i)osscsses  some  degree  of  elasticity,  and 
if  they  are  not  ke];t  together  after  the  imjnct  by  cohesion,  or  by  some 
artificial  appliance,  the  mutual  pressure  between  them  will  reach  a 
maximum,  will  begin  to  diminish,  and  m  the  end  will  come  to  nothing, 
by  gradually  diminishing  in  amount  per  unit  of  area  on  a  gr.iduariy 
diminishing  surface  of  coniact.  The  whole  process  would  >u:^-i:iy 
not  greatly  more  or  less  than  an  hour  if  the  bodies  were  of  siv.-Ii 
dimensions  as  the  earth,  and  such  degrees  of  rigidity  as  copper,  steel. 


86  PRELIMINARY. 

or  glass.  It  is  finished,  probably,  within  a  thousandth  of  a  second, 
if  they  are  globea  of  any  of  these  substances  not  exceeding  a  yard 
in  diameter. 

260.  The  whole  amount,  and  the  direction,  of  the  */w/a<r/' expe- 
rienced by  either  body  m  any  such  case,  are  reckoned  according  to 
'the  '  change  of  momentum'  which  it  experiences.  The  amount  of 
the  impact  is  measured  by  the  amount,  and  its  direction  by  the 
direction  of  the  change  of  momentum,  which  is  produced.  The 
component  of  an  impact  in  a  direction  parallel  to  any  fixed  line  is 
similarly  reckoned  according  to  the  component  change  of  momentum 
in  that  direction. 

261.  If  we  imagine  the  whole  time  of  an  impact  divided  into 
a  very  great  number  of  equal  intervals,  each  so  short  that  the  force 
does  not  vary  sensibly  during  it,  the  component  change  of  momentum 
in  any  direction  during  any  one  of  these  intervals  will  (§  185)  be 
equal  to  the  force  multiplied  by  the  measure  of  the  interval.  Hence 
the  component  of  the  impact  is  equal  to  the  sum  of  the  forces  in  all 
the  intervals,  multiplied  by  the  length  of  each  interval. 

262.  Any  force  in  a  constant  direction  acting  in  any  circumstances, 
for  any  time  great  or  small,  may  be  reckoned  on  the  same  principle ; 
so  that  what  we  may  call  its  whole  amount  during  any  time,  or  its 
'•  tinic-intcgral^  will  measure,  or  be  measured  by,  the  whole  momentum 
which  It  generates  in  the  time  in  question.  But  this  reckoning  is  not 
often  convenient  or  useful  except  when  the  whole  operation  con- 
sidered IS  over  before  the  position  of  the  body,  or  configuration  of 
the  system  of  bodies,  involved,  has  altered  to  such  a  .degree  as  to 
brmg  any  other  forces  into  play,  or  alter  forces  previously  acting, 
to  5uch  an  extent-  as  to  produce  any  sensible  effect  on  the  i^ipmentum 
measured.  Thus  if  a  person  presses  gently  with  his  hand,  during 
a  few  seconds,  upon  a  mass  suspended  by  a  cord  or  chain,  he  pro- 
duces an  effect  which,  if  we  know  the  degree  of  the  force  at  each 
instant,  may  be  thoroughly  calculated  on  elementary  principles.  No 
approximation  to  a  full  determination  of  the  motion,  or  to  answering 
such  a  partial  question  as  'how  great  will  be  the  whole  deflection 
produced?'  can  be  founded  on  a  knowledge  of  the  "■  tlmc-integrar 
alt)nc.  If,  for  mstance,  the  force  be  at  first  very  great  and  gradually 
diminish,  the  effect  will  be  very  different  from  what  it  would  be  if  the 
force  were  to  increase  very  gradually  and  to  cease  suddenly,  even 
although  the  time-integral  were  the  same  in  the  two  cases.  But  if 
the  same  body  is  '  struck  a  blow,'  in  a  horizontal  direction,  either  by 
the  hand,  or  by  a  mallet  or  other  somewhat  hard  mass,  the  action 
of  the  force  is  finished  before  the  suspending  cord  has  experienced 
any  sensible  deflection  from  the  vertical.  Neither  gravity  nor  any 
Other  force  sensibly  alters  the  effect  of  the  blow.  And  therefore  the 
whole  momentum  at  the  end  of  the  blow  is  sensibly  eciual  to  the 
'amount  of  the  impact,'  which  is,  in  this  case,  simply  the  time- 
integral. 


DYNAMICAL  LAWS  AXD  PRINCIPLES.  »; 

263.  Such  is  the  case  of  Robins'  Badistic  Pendulum,  a  massive 
block,  of  wood  movable  about  a  horizontal  axis  at  a  considerable 
distance  above  it — employed  to  measure  the  velocity  of  a  cannon  or 
musket-shot.  The  shot  is  fired  into  the  block  in  a  horizontal  direc- 
tion perpendicular  to  the  axis.  The  impulsive  pcnetrati(jn  is  so 
nearly  instantaneous,  and  the  inertia  of  the  block  so  hirge  compared 
wuh  the  momentum  of  the  shot,  that  the  ball  and  pendulum  are 
mo\'ini;  oi\  as  one  mass  bcf'Tc  t/if  penduIiDn  lias  ht:ai  snisiblx  drj/.\-f-\f 

/ran  the  p.^sition  of  equitibriuni.  This  is  the  essential  peculiarity  of  trie 
balhstic  method  ;  which  is  used  alao  extens!\'ely  in  electro-magnetic 
researches  and  in  practical  electric  testing,  when  the  integral  'luantity 
oftiie  elfinricity  whicli  has  passed  in  a  current  of  short  duratioii  is  to 
be  measured.  The  ballistic  formula  (§  272)  is  applicable,  with  tlie 
proper  change  of  notation,  to  all  such  cases. 

264.  Other  illustrations  of  the  cases  in  which  the  time-integral 
gi\-es  us  the  nimjilc-te  solution  of  the  problem  may  be  given  witliout 
limit.  They  include  all  cases  in  which  the  direction  of  the  force  is 
always  coincident  with  the  direction  of  motion  of  the  moving  body, 
and  those  speci:"d  rises  in  which  the  time  of  action  q{  the  force  is  so 
short  that  the  l)ody's  motion  does  not,  during  its  lapse,  sensibly  alter 
its  relation  to  the  direction  of  the  force,  or  the  action  of  any  other 
forces  to  which  it  may  be  subject.  Thus,  in  the  vertical  fall  of  a 
bod,-,  the  time-integral  gives  us  at  once  tlie  change  of  monicntum; 
and  the  same  rule  a;)plies  in  most  cases  of  forces  of  brief  duration, 
as  in  a  'drive'  in  cricket  or  golf 

265.  The  simplest  case  which  we  can  consider,  and  tl:c  one  usuiUy 
treated  as  an  introduction  to  the  subject,  is  that  of  the  collision  of 
two  smooth  spherical  bodies  whose  centres  before  collision  were 
muving  in  the  same  strrnglit  line.  The  force  between  them  at  each 
instant  mu>t  be  in  this  line,  becatise  of  the  symmetry  of  circums'au'ses 
round  it  ,  and  by  the  tinrd  law  it  nu!st  be  e^ual  in  amount  011  the 
two  bodies.  Hence  (Li:x  II.)  thev  must  experience  cliangcs  of 
motion  at  equal  rates  in  contrary  directions;  and  at  any  instant  of 
the  imjjact  the  integral  amotmts  of  these  changes  of  motion  must  be 
e<[ual.  Let  us  suppose,  to  fix  the  ideas,  the  two  bodies  to  be  moving 
both  before  and  after  impact  in  the  same  direction  in  one  line  .  one 
of  them  gaining  on  the  other  before  impact,  and  either  follo\vir,g  it 
at  a  less  s[)eed,  or  moving  along  witli  it,  as  tlie  case  may  l^e,  after 
the  impact  is  completed.  Cases  in  which  the  former  is  driven  back- 
wards by  the  force  of  the  collision,  or  in  which  the  two  moving  in 
opjKjsite  directions  meet  in  collision,  are  easily  reduced  to  dependence 
on  the  same  fjrnnila  by  t!ie  ordinar}-  algebraic  convention  with  regard 
to  positi\e  and  negative  signs. 

In  the  standard  case,  then,  the  quantity  of  motion  lost,  up  to  any 
instant  of  the  impact,  by  one  of  the  bodies,  is  equal  tu  that  gained 
by  the  other.  Hence  at  tb.e  instant  wjien  their  velocities  are  equaiiz'-d 
they  move  as  one  mass  with  a  momentum  equal  to  the  sum  of  the 


88  FRELIMINARY. 

momenta  of  the  two  before  impact,     That  is  to  say,  if  v  denote  the 
common  velocity  at  this  instant,*  we  have 

( J/+  M')  V  =  MF+  M'  r, 
''  MV^M'V 
M  +  M 

if  M,  M'  denote  the  masses  of  the  .two   bodies,  and    V,    V  their 
velocities  before  impact. 

During  this  fi,rst  period  of  the  impact  the  bodies  have  been,  on 
the  whole,  coming  jnto  closer  contact  with  one  another,  through  a 
compression  or  deformation  experienced  by  6ach,  and  .resultmg,  as 
remarked  above,  in  a  fitting  together  of  the  two  surfaces  over  a 
finite  area.  No  body  in  nature  is  perfectly  inelastic ;  and  hence, 
at  the  instant  of  closest  approximation,  the  mutual  force  called 
into  action  between  the  two  bodies  continues,  and  tends  to  separate 
them.  Unless  prevented  by  natural  surface  cohesion  or  welding  (such 
as  is  always  found,  as  we  shall  see  later  in  our  chapter  on  Properties 
of  Matter,  however  hard. and  well  polished  the  surfaces  may  be),  or 
by  artificial  appliances  (such  as  a -coating  of  wax,  applied  in  one  of 
the  common  illustrative  experiments;  or  the  coupling  applied  between 
two  railway-carriages  when  run  together  so  as  to  push  in  the  springs, 
according  to  the  usual  practice  at  railway-stations),  the  two  bodies  are 
actually  separated  by  this  force,  and  move  away  from  one  another. 
Newton  found  XhsX,  provided  the  impact  is  not  so  violc?it  as  to  make  any 
sensible  permanent  indentation  in  either  body,  the  relative  velocity  of 
separation  after  the  impact  bears  a  proportion  to  their  previous 
relative  velocity  of  approach,  which  is  constant  for  the  same  two 
bodies.  This  proportion,  always  less  than  unity,  approaches  more 
and  more  nearly  to  it  the  harder  the  bodies  are.  Thus  with  balls  of 
compressed  wool  he  found  it  |,  iron  nearly  the  same,  glass  ^|.  The 
results  of  more  recent  experiments  on  the  same  subject  have  con- 
firmed Newton'5  law.  These  will  be  described  later.  In  any  case 
of  the  collision  of  tv/o  balFs,  let  e  denote  this  proportion,  to  which  we 
give  the  name  Coefficient  of  Restitution'' ;  and,  with  previous  nota- 
tion, let  in  addition  U,  U'  denote  the  velocities  of  the  two  bodies 
after  the  conclusion  of  the  impact ;  in  the  standard  case  each  beinj 
positive,  but  U'  >  U.     Then  we  have 

U' -  U=e^V-  V), 
and,  as  before,  since  one  has  lost  as  much  momentum  as  the  other 
has  gained,  MU+  M'  U'  =  MV+  M'  V. 

From  these  equations  we  find 

{M^  M')U=  AfV+  M'  V  -  eAI'  (  V-  V'\ 
with  a  similar  expression  for  U' . 

1  In  most  modern  lre?.tises  this  is  called  a  'coefficient  of  elasticity;'  a 
misnomer,  suggested,  it  may  be,  by  Newton's  words,  but  utterly  at  variance  with 
modern  language  and  modern  knowledge  regarding  elasticity. 


DYXAMICAL   LAIVS  AND    PRINCIPLES.  89 

Also  we  have,  as  above, 

(.)/+  ^f)  V  =  MV-^  M  v. 
Hence,  by  subtraction, 

[M  +  M')  {V  -U)^  eAP  (  K-  n  =  e\.\r  V-  (J/-  J/')  :•  ■.  MV\, 
and  therefore  v  -  U=  e  (  V-  v). 

Of  course  we  have  also  If  -v  -e  {:/  -  F'). 

These  results  may  be  put  in  words  thus : — The  relativf  velocity  of 
either  of  the  bodies  with  regard  to  the  centre  of  inertia  of  the  two 
IS,  after  the'  completion  of  the  impact,,  reversed  in  direction,  ancf 
diminished  in  the  ratio  e  :  i. 

266.  Hence  the  loss  of  kinetic  energy,  being,  according  to  §§  233, 
234,  due  only  to  change  of  kinetic  energy  relative  to  the  o  itrc  of 
inertia,  is  to  this  part  of  the  whole  as  i  -  e^ :  i- 

Thus  by  §  234, 
Initial  kmetic  energ)-  =  \  {M+Jr)t^  +  Uf  ( T-  ^0'  +  i  -'^  {'-'-  H'- 
Fmal        „        ,„       =l{M+M'}v'  +  ^^JI{v-[/y  +  lAr{(/'-zf 

Loss  =  1  ( I  -  ^)  {j/(  V-  vy  +  M'  {v  -  vy\. 

267.  When  two  elastic  bodies,  the  two  balls  supposed  above  for 
instance,  impinge,  some  portion  of  their  previous  kinetic  energy  will 
always  remain  in  them  as  vibrations.  K  pcrtion  of  the  loss  of  energy 
(miscalled  the  effect  of  imperfect  elasticity  alone)  is  necessarily  duo 
to  this  cause  in  every  real  case.  _ 

Later,  in  our  chapter  on  the  Properties  of  Matter,  it  will  be  shown 
as  a  result  of  experiment,  that  forces  of  elasticity  are,  to  a  very  close 
degree  of  accuracy,  simply  proponional  to  the  strains  (§  135),  within 
the  limits  of  elasticity,  in  elastic  solids  which,  like  metals,  glass,  etc., 
bear  but  small  deformations  without  permanent  change.  Hence  when 
two  such  bodies  come  mto  coUision-,  sometimes  wit^  greater  and 
sometimes  with  less  mutual  velocity,  but  with  all  other  circumstances 
similar,  the  velocities  of  all  particles  of  either  body,  at  corresponding 
times  of  the  impacts,  will  be  always  in  the  same  proportion.  Hence 
the  velocity  of  separation  of  the  centres  of  inertia  after  impact  will 
bear  a  constant  proportion  to  the  previous  velocity  of  approach ; 
which  agrees  with  the  Newtonian  law.  It  is  therefore  probable  that 
a  very  sensible  portion,  if  not  the  whole,  of  the  loss  of  energ}'  m  the 
visible  motions  of  two  elastic  bodies,  after  impact,  experimented  on 
by  Newton,  may  have  been  due  to  vibrations ;  but  unless  some  other 
cause  also  was  largely  operative,  it  is  difficult  to  see  how  the  loss  was 
so  much  greater  with  iron  balls  than  with  glass. 

268.  In  certain  definite  extreme  cases,  imaginable  although  not 
realizable,  no  energy  will  be  spent  in  vibrations,  and  the  two  bodies 
will  separate,  each  moving  simply  as  a  rigid  body,  and  having  in  thh 
simple  motion  the  whole  energy  of  work  done  on  it  by  elastic  force 
during  the  collision.  For  instance,  let  the  two  bodies  be  cylinders, 
or  prisnutic  bars  with  flat  ends,  of  the  same  kind  of  substance,  and  of 

Yol.  23—5 


90  PRELUflNARY. 

equal  and  similar  transverse  sections ;  and  let  this  substance  have  the 
property  of  cortTpressibility  wth  perfect  elasticity,  in  the  direction  of 
tlic  length  of  the  bar,  and  of  absolute  resistance  to  change  in  every 
transverse  dimension.  Before  impact,  let  the  \.\<o  bodies  be  placed 
\yiih  iheir  lengths  in  one  line,  and  their  transverse  sections  (if  not 
circular)  similarly  situated,  and  let  one  or  both  be  set  in  mouon  in 
this  line.  Then,  if  the  lengths  of  the  two  be  equal,  they  will  separate 
after  impact  with  the  same  relative  velocity  as  that  with  which  they 
approached,  and  neither  will  retain  any  vibratory  motion  after  the 
end  of  the  collision.  The  result,  as  regards  the  motions  of  the  two 
bodies  after  the  collision,  -will  be  sensibly  the  same  if  they  are  of  any 
real  ordinary  elastic  solid  material,  provided  the  greatest  transverse 
diameter  of  each  is  very  small  in  comparison  of  its  length. 

289.    If  the  two  bars  are  of  an  unequal  length,  the  shorter  will,  after 

the  impact,  be  in  exactly  the  same  state  as  if  it  had  struck  another 

of  its  own  length,  and  it  therefore  will  move  as  a  rigid  body  after  the 

collision.     But  the  other  will,  along  with  a  motion  of  its  centre  of 

gravity,  calculable  from  the  principle  that  its  whole  momentum  must 

(§  ^33)  be  changed  by  an  amount  equal  exactly  to  the  momentum 

gained  or  lost  by  the  first,  have  also  a  vibratory  motion,  of  which  the 

whole  kinetic  and  potential  energy  will  make  up  the  deficiency  of 

energy  which  we  shall  presently  calculate  in  the  motions  of  the  centres 

of  inertia.     For  simplicity,  let  the  longer  body  be  supposed  to  be  at 

rest  before  the  collision.     Then  the  shorter  on  striking  it  will  be  left 

at  rest;  this  being  clearly  the  result  in  the  case  of  the  e  —  \  in  the 

preceding  formulae  (§  265)  applied  to  the  impact  of  one  body  striking 

another  of  equal  mass  previously  at  rest.     The  longer  bar  will  move 

away  with  the  same  momentum,  and  therefore  with  less  velocity  of  its 

centre  of  inertia,  and  less  kinetic  energy  of  this  motion,  than  the  other 

body  had  before  impact,  in  the  ratio  of  the  smaller  to  the  greater 

mass.     It  will  also  have  a  very  remarkable  vibratory  motion,  which, 

when  its  length  is  more  than  double  of  that  of  the  other,  will  consist 

of  a  wave  running  backwards  and  forwards  through  its  length,  and 

causing  the  motion  of  its  ends,  and,  in  fact,  of  every  particle  of  it,  to 

take  place  by  'fits  and  starts,'  not  continuously.     The  full  analysis  of 

these  circumstances,  though  very  simple,  must  be  reserved  until  we 

are  especially  occupied  with  waves,  and  the  jcinetics  of  elastic  solids. 

It  IS  sufiicicnt  at  present  to  remark,  that  the  motions  of  the  centres  of 

inertia  of  the  two  bodies  after  impact,  whatever  they  may  have  been 

previously,  are  given  by  the  preceding  formulae  with  for  e  the  value 

M' 

-jjr ,  where  J/ and  M'  are  the  smaller  and  larger  mass  respectively. 

270.  The  mathematical  theory  of  the  vibrations  of  solid  elastic 
spheres  has  not  yet  been  worked  out ;  and  its  application  to  the  case 
of  the  vibrations  produced  by  impact  presents  considerable  difficulty. 
Experiment,  however,  renders  it  certain,  that  but  a  small  part  of  the 
•whole  kinetic  energy  of  the  previous  motions  can  remain  in  the  form 
of  vibrations  after  the  impact  of  two  equal  spheres  of  glass  or  of 


DYNAMICAL  LAWS  AND  PRINCIPLES.  91 

ivory.  This  is  proved,  foit  instance,  by  the  common  observation,  that 
ovtK.  of  them  remains  nearly  motionless  after  striking  the  other  pre- 
viously at  rest;  since,  the  velocity  of  the  common  centre  of  inertia  of 
the  two  being  necessarily  unchanged  by  the  impact,  we  infer  that  the 
second  ball  acquires  a  velocity  nearly  equal  to  that  which  the  first  had 
before  striking  it.  But  it  is  to  be  expected  that  unequal  balls  of  the 
same  substance  coming  into  collision  will,  by  impact,  convert  a  very 
sensible  proportion  of  the  kinetic  energy  of  their  previous  motions 
into  energy  of  vibrations ;  and  generally,  that  the  same  will  be  the 
case  when  equal  or  unequal  masses  of  different  substances  come  into 
collision ;  although  for  one  particular  proportion  of  their  diameters, 
depending  on  their  densities  and  elastic  qualities,  this  effect  will  be 
a  minimum,  and  possibly  not  much  more  sensible  than  it  is  when  the 
substances  are  the  same  and  the  diameters  equal. 

271.  It  need  scarcely  be  said  that  in  such  cases  of  impact  as  that 
of  the  tongue  of  a  bell,  or  of  a  clock-hammer  striking  its  bell  (or 
spiral  spring  as  in  the  American  clocks),  or  of  pianoforte-hammers 
striking  the  strings,  or  of  the  drum  struck  with  the  proper  implement, 
a  large  part  of  the  kinetic  energy  of  the  blow  is  spent  in  generating 
vibrations. 

272.  The  Moment  of  an  Impact  about  any  axis  is  derived  from  the 
line  and  amount  of  the  impact  in  the  same  way  as  the  moment  of 
a  velocity  or  force  is  determined  from  the  line  and  amount  of  the 
velocity  or  force,  §  46.  If  a  body  is  struck,  the  change  of  its 
moment  of  momentum  about  any  axis  is  equal  to  the  moment  of  the 
impact  round  that  axis.  But,  without  considering  the  measure  of  the 
impact,  we  see  (§  233)  that  the  moment  of  momentum  round  any  axis, 
lost  by  one  body  in  striking  another,  is,  as  in  every  case  of  mutual 
action,  equal  to  that  gained  by  the  other. 

Thus,  to  recur  to  the  ballistic  pendulum — the  line  of  motion  oi  the 
bullet  at  impact  may  be  in  any. direction  whatever,  but  the  only  part 
which  is  effective  is  the  component  in  a  plane  perpendicular  to  the 
axis.  We  may  therefore,  for  simplicity,  consider  the  motion  to  be  in 
a  line  perpendicular  to  the  axis,  though  not  necessarily  horizontal. 
Let  ;;;  be  the  mass  of  the  bullet,  v  its  velocity,  and  p  the  distance  of 
its  hne  of  motion  from  the  axis.  Let  M  be  the  mass  of  the  pendulum 
with  the  bullet  lodged  in  it,  and  k  its  radius  of  gyration.  Then  if  a> 
be  the  angular  velocity  of  the  pendulum  when  the  impact  is  complete, 

mvp  =  Mk^ui, 
iTom  which  the  solution  of  the  question  is  easily  determined. 

For  the  kinetic  energy  after  impact  is  changed  (§  207)  into  its 
equivalent  in  potential  energy  when  the  pendulum  reaches  its  position 
of  greatest  deflection.  Let  this  be  given  by  the  angle  0 :  then  the 
height  to  which  the  centre  of  inertia  is  raised  is  /i  (i  -  cos  $)  if  A  be  its 
distance  from  the  axis.     Thus 


Mg/i  ( I  -  cos  ^)  =  iiMkW  =  i  ^^' 


Mk* 


92  PRELIMINARY. 

.6     m   p      V        m    p   Ttv    ..  „,  fk^ 

or         2  sm  -  =  •?>.  t:  .  --^^-  :=  -,-„ .  ^,  .  -^., ,  :f  T~it    /  -- 

2     y>/  k    ^^/i     M  h  gT^  V  gf"- 

an  expression  for  the  chord  of  the  angle  of  deflection.  In  practice 
the  chord  of  the  angle  d  is  measured  by  means  of  a  light  tape  or 
cord  attached  to  a  point  of  the  pendulum,  and  slipping  with  small 
friction  through  a  clip  fixed  close  to  the  position  occupied  by  that 
point  when  the  pendulum  hangs  at  rest. 

273.  Work  dotie  by  an  impact  is,  in  general,  the  product  of  the 
impact  into  half  the  sum  of  the  initial  and  final  velocities  of  the  point 
at  which  it  is  applied,  resolved  in  the  direction  of  the  impact ,  In  the 
case  of  direct  impact,  such  as  that  treated  in  §  265,  the  initial  kinetic 
energy  of  the  body  is  \MV^i  the  final  \MU'^^  and  therefore  the  gain 
By  the  igipact  is 

\M{U"-~V\ 
or,  Yi'hich  is  the  same, 

MiJJ-  V).\{U^  V). 
But  M{U~  V)  is  (§  260)  equal  to  the  amount  of  the  impact.    Hence 
the  proposition :    the   extension  of  which  to  the  most  general  cir- 
cumstances is  not  difficult,  but  requires  somewhat  higher  ana^sis 
than  can  be  admitted  here. 

274.  It  is  worthy  of  remark,  that  if  any  number  of  im])acts  be 
applied  to  a  body,  their  whole  effect  will  be  the  same  whether  they 
be  applied  together  or  successively  (provided  that  the  whole  time 
occupied  by  them  be  infinitely  short),  although  the  work  done  by 
each  particular  impact  is,  in  general,  different  according  to  the  order 
in  which  the  several  impacts  are  applied.  The  whole  amount  of 
work  is  the  sum  of  the  products  obtained  by  multiplying  each  impact 
by  half  the  sum  of  the  components  of  the  initial  and  final  velocities 
of  the  point  to  which  it  is  applied? 

275.  The  effect  of  any  stated  impulses,  applied  to  a  rigid  body, 
or  to  a  system  of  material  points  or  rigid  bodies  connected  in  any 
way,  is  to  be  found  most  readily  by  the  aid  of  D'Alembert's  principle; 
according  to  v^hich  the  given  impulses,  and  the  impulsive  reaction 
against  the  generation  of  motion,  measured  in  amount  by  the 
mom.enta  generated,  are  in  equilibrium;  and  are,  therefore,  to  be 
dealt  with  mathematically  by  applying  to  -them  the  equations  of 
equilibrium  of  the  system. 

276.  [A  material  system  of  any  kind,  given  at  rest,  and  subjected 
to  an  impulse  in  any  specified  direction,  and  of  any  given  magnitude, 
moves  off  so  as  to  take  the  greatest  amount  of  kinetic  energy  which 
the  specified  impulse  can  give  it. 

277.  If  the  system  is  guided  to  takej  under  the  action  of  a  given 
impulse,  any  motion  different  from  the  natural  motion,  it  will  have 
less  kinetic  energy  than  that  of  the  natural  motion,  by  a  difference 
equal  to  the  kinetic  energy  of  the  motion  represented  by  the  resultant 
(§  67)  of  those  two  motions,  one  of  them  reversed. 


DYNAMICAL  LAWS  AND  PRINCIPLES.  93 

Cor.  If  a  set  of  material  points  are  struck  independently  by 
impulses  each  given  m  amount,  more  kinetic  energy  is  generated  \i 
the  points  are  perfectly  free  to  move  each  mdepeaJently  of  all  the 
others,  than  if  they  are  connected  m  any  way.  And  the  deficiency 
of  energy  in  the  latter  case  is  equal  to  the  amoun:  of  the  kinetic 
energy  of  the  motion  which  geometrically  compounded  with  the 
motion  of  either  case  would  give  that  of  the  other. 

.  278.  Given  any  material  system  at  rest.  Let  any  parts  of  it  be 
set  in  motion  suddenly  with  any  specified  velocities,  possible  accord- 
ing to  the  conditions  of  the  system ;  and  let  its  other  parts  be 
influenced  only  by  its  connexions  with  those.  It  is  required  to 
find  the  motion.  The  solution  of  the  problem  is — The  motion 
actually  taken  by  the  system  is  that  which  has  less  kinetic  energ}'than 
any  other  motion  fulfilling  the  prescribed  velocity  conditions.  And 
the  e.xcess  of  the  energy  of  any  other  such  motion,  above  that  of  the 
actual  motion,  is  equal  to  the  energy  of  the  motion  that  would  be 
generated  by  the  action  alone  of  the  impulse  which,  if  compounded 
with  the  impulse  producing  the  actual  motion,  would  produce  this 
other  supposed  motion.] 

279.  Maupertuls'  celebrated  principle  of  Least  Action  has  been, 
even  up  to  the  present  time,  regarded  rather  as  a  curious  and  some- 
what perplexing  property  of  motion,  tha-n  as  a  useful  guide  in  kinetic 
investigations.  U'e  are  strongly  impressed  with  the  conviction  that 
a  much  more  profound  importance  will  be  attached  to  it,  not  only 
in  abstract  dynamics,  but  in  the  theory  of  the  several  branches  of 
physical  science  now  beginning  to  receive  dynamic  explanations- 
As  an  extension  of  it.  Sir  W.  R.  Hamilton'  has  evolved  his  method 
of  Varying  Action.,  which  undoubtedly  must  become  a  most  valuable 
aid  in  luture  generalizations. 

What  IS  meant  by  'Action'  in  these  expressions  is,  unfortunately, 
something  very  different  from  the  Actio  Agentis  defined  by  Newton, 
and,  it  must  .be  admitted,  is  a  much  less  judiciously  chosen  word. 
Taking  it,  however,  as  we  find  it  now  universally  used  by  writers  on 
dynamics,  we  define  the  Action  of  a  Moving  System  as  proportional 
to  the  average  kmetic  energy,  which  it  has  possessed  during  tlie  time 
from  any  convenient  epoch  of  reckoning,  multiplied  by  the  time. 
According  to  the  unit  generally  adopted,  the  action  of  a  system 
which  has  not  varied  in  its  kinetic  energy,  is  twice  the  amount  of  the 
energy  multiplied  by  the  time  from  the  epoch.  Or  if  the  energy  has 
been  sometimes  greater  and  sometimes  less,  the  action  at  time  / 
is  the  double  of  what  we  may  call  the  time-integral  of  the  energy; 
that  IS  to  say,  the  action  of  a  system  is  equal  to  the  sum  of  the 
average  momenta  /or  the  spaces  described  by  the  particles  from  any 
era  each  multiplied  by  the  length  of  us  path 

280.  The  principle  of  Least  Action  is  this  . — Of  all  the  different 
sets  of  paths  along  which  a  conservative  system  may  be  guided  to 
move  from  one  configuration  to  another,  with  the  sum  of  us  puicntial 

'  Phil.  Trans.,  1834— 1835. 


94  PRELIMINARY. 

and  kinetic  energies  equal  to  a  given  constant,  that  one  for  which  the 
action  is  the  least  is  such  that  the  system  will  require  only  to  be 
started  with  the  proper  velocities,  to  move  along  it  unguided. 

281.  [In  any  unguided  motion  whatever,  of  a  conservative  system, 
the  Action  from  any  one  stated  position  to  any  other,  though  not 
necessarily  a  minimum,  fulfils  the  statto?iary  condition,  that  is  to  say, 
the  condition  that  the  variation  vanishes,  which  secures  either  a 
minimum  or  maximum,  or  maximum-minimum.] 

282.  From  this  principle  of  stationary  action,  founded,  as  we  have 
seen,  on  a  comparison  between  a  natural  motion,  and  any  other 
motion,  arbitrarily  guided  and  subject  only  to  the  law  of  energy,  the 
initial  and  final  configurations  of  the  system  being  the  same  in  each 
case;  Hamilton  passes  to  the  consideration  of  the  variation  of  the 
action  in  a  natural  or  unguided  motion  of  the  system  produced  by 
varying  the  initial  and  final  configurations,  and  the  sum  of  the 
potential  and  kinetic  energies.     The  result  is,  that — 

283.  The  rate  of  decrease  of  the  action  per  unit  of  increase  of 
any  one  of  the  free  (generalized)  co-ordinates  specifying  the 
initial  configuration,  is  equal  to  the  corresponding  (generalized)  com- 
ponent momentum  of  the  actual  motion  from  that  configuration : 
the  rate  of  increase  of  the  action  per  unit  increase  of  any  one 
of  the  free  co-ordinates  specifying  the  final  configuration,  is  equal 
to  the  corresponding  component  momentum  of  the  actual  motion 
towards  this  second  configuration:  and  the  rate  of  increase  of  the 
action  per  unit  increase  of  the  constant  sum  of  the  potential  and 
kinetic  energies,  is  equal  to  the  time  occupied  by  the  motion  of 
wlaich  the  action  is  reckoned. 

284.  The  determination  of  the  motion  of  any  conservative  system 
from  one  to  another  of  any  two  configurations,  when  the  sum  of  its 
potential  and  kinetic  energies  is  given,  depends  on  the  determination 
of  a  single  function  of  the  co-ordinates  specifying  those  configura- 
tions by  means  of  two  quadratic,  partial  differential  equations  of  the 
first  order,  with  reference  to  those  two  sets  of  co-ordinates  respec- 
tively, with  the  condition  that  the  corresponding  terms  of  the  two 
differential  equations  become  separately  equal  when  the  values  of 
the  two  sets  of  co-ordinates  agree.  The  function  thus  determined 
and  em,^Ioyed  to  express  the  solution  of  the  kinetic  problem  was 
called  the  Characteristic  Function,  by  Sir  W.  R.  Hamilton,  to  whom 
the  method  is  due.  It  is,  as  we  have  seen,  'the  'action'  from  one 
of  the  configurations  to  the  other;  but  its  peculiarity  in  Hamilton's 
system  is,  that  it  is  to  be  expressed  as  a  function  of  the  co-ordinates 
and  a  constant,  the  whole  energy,  as  explained  above.  It  is  evidently 
symmetrical  with  respect  to  the  two  configurations,  changing  only  in 
sign  if  their  co-ordinates  are  interchanged. 

285.  The  most  general  possible  solution  of  the  quadratic,  partial 
differential  equation  of  the  first  order,  satisfied  by  Hamilton's  Qha- 


DYNAMICAL  LAWS  AND  PRINCIPLES.  95 

racteristic  Function  (either  tenninal  configuration  alone  varying), 
when  interpreted  for  the  case  of  a  single  free  particle,  expresses  the 
action  up  to  any  point  from  some  point  of  a  certam  arbitrarily 
given  surface,  from  which  the  particle  has  been  projected,  in  the 
direction  of  the  normal,  and  with  the  proper  velocity  to  make  the 
sum  of  the  potential  and  actual  energies  have  a  given  value.  In  other 
words,  the  physical  problem  solved  by  the  most  general  solution  of 
that  partial  differential  equation,  for  a  single  free  particle,  is  this: — 

Let  free  particles,  not  mutually  influencing  one  another,  be  pro- 
jected normally  from  all  points  of  a  certain  arbitrarily  given  surface, 
each  with  the  proper  velocity  to  make  the  sum  of  its  potential  and 
kinetic  energies  have  a  given  value.  To  find,  for  that  one  of  the 
particles  which  passes  through  a  given  point,  the  'action' in  its  course 
from  the  surface  of  projection  to  this  point.  The  Hamiltonian 
principles  stated  above,  show  that  the  surfaces  of  equal  action  cut 
the  paths  of  the  particles  at  right  angles;  and  give  also  the 
following  remarkable  properties  of  the  motion : — 

If,  from  all  points  of  an  arbitrary  surface,  particles  not  mutually 
influencing  one  another  be  projected  with  the  proper  velocities  in 
the  directions  of  the  normals;  points  which  they  reach  with  equal 
actions  he  on  a  surface  cutting  the  paths  at  right  angles.  The 
infinitely  small  thickness  of  the  space  between  any  two  such  surfaces 
corresponding  to  amounts  of  action  differing  by  any  infinitely  small 
quantity,  is  inversely  proportional  to  the  velocity  of  the  particle 
traversing  it ;  being  equal  to  the  infinitely  small  difference  of  action 
divided  by  the  whole  momentum  of  the  particle. 

288.  Irrespectively  of  methods  for  finding  the  'characteristic 
function'  in  kinetic  problems,  the  fact  that  any  case  of  motion  what- 
ever can  be  represented  by  means  of  a  single  function  in  the  manner 
explained  in  §  284,  is  most  remarkable,  and,  when  geometrically 
interpreted,  leads  to  highly  important  and  interesting  properties  of 
motion,  which  have  valuable  applications  in  various  branches  of 
Natural  Philosophy ;  ooe  of  which,  explained  below,  led  Hamilton  ' 
to  a  general  theory  of  optical  instruments,  comprehending  the  whole 
in  one  expression.  Some  of  the  most  direct  applications  of  the 
general  principle  to  the  motions  of  planets,  comets,  etc.,  considered 
as  free  points,  and  to  the  celebrated  problem  of  perturbations,  known 
as  the  Problem  of  Three  Bodies,  are  worked  out  in  considerable  detail 
by  Hamilton  {Phil.  Trans.,  1834-5),  and  in  various  memoirs  by 
Jacobi,  Liouville,  Bour,  Donkin,.Cayley,  Boole,  etc. 

The  now  abandoned,  but  still  interesting,  corpuscular  theory  of 
light  furnishes  the  most  convenient  language  for  expressing  the 
optical  application.  In  this  theory  light  is  supposed  to  consist  of 
material  particles  not  mutually  influencing  one  another ;  but  subject 
to  molecular  forces  from  the  particles  of  bodies,  not  sensible  at 
sensible  distances,  and  therefore  not  causing  any  deviation  from 
uniform  rectilinear  motion  in  a  homogeneous  medium,  except  within 

*  On  tht  Theory  of  Syuems  of  Rays.     Trans.  R.  1.  A..  18J4.  «83o,  i8jj. 


96  PRELIMINARY. 

an  indefinitely  small  distance  from  its  boundary.  The  laws  of  reflec- 
tion and  of  single  refraction  follow  correctly  from  this  hypothesis, 
which  therefore  suffices  for  what  is  called  geometrical  optics. 

We  hope  to  return  to  this  subject,  with  sufficient  detail,  in  treating 
of  Optics.  At  present  we  limit  ourselves  to  state  a  theorem  com. 
prehending  the  known  rule  for  measuring  the  magnifying  power  of 
•a  telescope  or  microscope  (by  comparing  the  diameter  of  the  objects 
glass  with  the  diameter  of  pencil  of  parallel  rays  emerging  from  the 
eye-piece,  when  a  point  of  light  is  placed  at  a  great  distance  in  front 
of  the  object-glass),  as  a  particular  case. 

287.  Let  any  number  of  attracting  or  repelling  masses,  or  perfectly 
smooth  elastic  objects,  be  fixed  in  space.  Let  two  stations,  (9  and  0\ 
be  chosen.  Let  a  shot  be  fired  with  a  stated  velocity,  F,  from  O, 
in  such  a  direction  as  to  pass  through  O'.  There  may  clearly  be 
more  than  one  natural  path  by  which  this  may  be  done;  but,  generally 
speaking,  when  one  such  path  is  chosen,  no  other,  not  sensibly  di- 
verging from  it,  can  be  found;  and  any  infinitely  small  deviation  in 
the  hne  of  fire  from  (9,  will  cause  the  bullet  to  pass  infinitely  near  to, 
but  not  through  O'  Now  let  a  circle,  with  infinitely  small  radius  r, 
be  described  round  O  as  centre,  in  a  plane  perpendicular  to  the  line 
of  fire  from  this  point,  and  let — all  with  infinitely  nearly  the  same 
velocity,  but  fulfilling  the  condition  that  the  sum  of  the  potential  and 
kinetic  energies  is  the  same  as  that  of  the  shot  from  O — bullets  be 
fired  from  all  points  of  this  circle,  all  directed  infinitely  nearly  parallel 
to  the  hne  of  fire  from  O,  but  each  precisely  so  as  to  pass  through  O' . 
Let  a  target  be  held  at  an  infinitely  small  distance,  a ,  beyond  0\ 
in  a  plane  perpendicular  to  the  line  of  the  shot  reaching  it  from  O. 
The  bullets  fired  from  the  circumference  of  the  circle  round  O,  will, 
after  passing  through  0\  strike  this  target  in  the  circumference  of  an 
exceedingly  small  ellipse,  each  with  a  velocity  (corresponding  of 
course  to  its  position,  under  the  law  of  energy)  differing  infinitely 
little  from  V,  the  common  velocity  with  which  they  pass  through  C/ 
Let  now  a  circle,  equal  to  the  former,  be  described  round  0\  in  the 
plane  perpendicular  to  the  central  path  through  O' ,  and  let  bullets  be 
fired  from  points  in  its  circumference,  each  with  the  proper  velocity, 
and  in  such  a  direction  infinitely  nearly  parallel  to  the  central  path 
as  to  make  it  pass  through  O.     These  bullets,  if  a  target  is  held  to 

y 
receive  them  perpendicularly  at  a  distance  a  =  a'  ~,  beyond  O,  will 

strike  it  along  the  circumference  of  an  ellipse  equal  to  the  former 
and  placed  in  a  corresponding  position;  and  the  points  struck  by  the 
individual  bullets  will  correspond  in  the  manner  explained  below. 
Let  P  and  P'  be  points  of  the  first  and  second  circles,  and  Q  and 
^  the  points  on  the  first  and  second  targets  which  bullets  from  them 
strike ;  then  if  P'  be  in  a  plane  containing  the  central  path  through 
O,  and  the  position  which  Q  would  take  if  its  ellipse  were  made 
circular  by  a  pure  strain  (§  159) ;  Q  and  Q  are  similarly  situated  on 
the  two  ellipses. 


DYNAMICAL  LAWS  AND  PRINCIPLES.  qj 

288.  The  most  obvious  optical  application  of  this  remarkable 
result  is,  that  in  the  use  of  any  optical  apparatus  whatever,  if  ths  eye 
and  the  object  be  interchanged  without  altering  the  position  of  the 
instrument,  tlie  magnifying  power  is  unaltered.  This  13  easily  under- 
stood when,  as  in  an  ordinary  telescope,  microscope,  cr  opera-glass 
((jalilean  telescope),  the  instrument  is  symmetrical  about  £in  a>ds,  and 
is  curiously  contradictory  of  the  common  idea  that  a  telescope  'dimi- 
nishes' when  looked  through  the  wrong  way,  which  no  doubt  is  true 
if  the  telescope  is  simply  reversed  about  the  middle  of  its  length,  eye 
and  object  remammg  hxed.  But  if  the  telescope  be  removed  from  the 
eye  till  its  eye-piece  is  close  to  the  object,  the  part  of  the  object  seen 
will  be  seen  enlarged  to  the  same  extent  as  when  viewed  with  the 
telescope  held  in  the  usual  manner.  This  is  easily  verified  by  looking 
from  a  distance  of  a  few  yards,  in  through  the  object-glass  of  an 
opera-glass,  at  the  eye  of  another  person  holding  it  to  his  eye  in 
the  usual  way. 

The  more  general  application  may  be  illustrated  thus: — Let  the 
points,  O,  O'  (the  centres  of  the  two  circles  described  in  the  preceding 
enunciation),  be  the  optic  centres  of  the  eyes  of  two  persons  looking 
at  one  another  through  any  set  of  lenses,  prisms,  or  transparent 
media  arranged  in  any  way  between  them.  If  their  pupils  are  of 
equal  sizes  in  nTnlity,  they  will  be  seen  as  similar  ellipses  of  equal 
apparent  dimensions  by  the  two  observers.  Here  the  imagined 
particles  of  light,  projected  from  the  circumference  of  the  pupil  of 
either  eye,  are  substituted  for  the  projectiles  from  the  circumference 
of  either  circle,  and  the  retina  of  the  other  eye  takes  the  place  of  the 
target  receiving  them,  in  the  general  kinetic  statement.      "^ 

289.  If  instead  of  one  free  particle  we  have  a  conservative  system 
of  any  number  of  mutually  influencing  free  particles,  the  same  state- 
ment may  be  apphed  with  reference  to  the  initial  position  of  one  ot 
the  particles  and  the  final  position  of  another,  or  with  reference  to  the 
initial  positions,  or  to  the  final  positions,  of  two  of  the  particles.  It 
thus  serves  to  show  how  the  influence  of  an  infinitely  small  change  in 
one  of  those  positions,  on  the  direction  of  the  other  particle  passing 
through  the  other  position,  is  related  to  the  influence  on  the  direction 
of  the  former  particle  passing  through  the  former  position  produced 
by  an  infinitely  small  change  in  the  latter  position,  and  is  of  immense 
use  in  physical  astronomy.  A  corresponding  statement,  in  terms  of 
generalized  co-ordinates,  may  of  course  be  adapted'  to  a  system  of 
rigid  bodies  or  particles  connected  in  any  way.  All  such  statements 
are  included  in  the  following  very  general  proposition: — 

The  rate  of  increase  of  the  component  momentum  relative  to 
any  one  of  the  co-ordinates,  per  unit  of  increase  of  any  other  co- 
ordinate, is  equal  to  the  rate  of  increase  of  the  component  momentum 
relative  to  the  latter  per  unit  increase  or  diminution  of  the, former 
co-ordinate,  according  as  the  two  co-ordinates  chosen  belong  to  one 
configuration  of  t-he  system,  or  one  of  them  belongs  to  the  initial 
configuration  and  the  other  to  the  final. 


98  PRELIMINARY. 

290.  If  a  conservative  system  is  infinitely  little  displaced  from  a 
configuration  of  stable  equilibrium,  it  will  ever  after  vibrate  about  this 
configuration,  remaining  infinitely  near  it;  each  particle  of  the  system 
performing  a  motion  which  is  composed  of  simple  harmonic  vibra- 
tions. If  there  are  /  degrees  of  freedom  to  move,  and  we  consider 
any  system  of  generalized  co-ordinates  specifying  its  position  at 
any  time,  the  deviation  of  any  one  of  these  co-ordinates  from  its 
value  for  the  configuration  of  equilibrium  will  vary  according  to  a 
complex  harmonic  function  (§  88),  composed  in  general  of  /  simple 
harmonics  of  incommensurable  periods,  and  therefore  (§  85)  the  whole 
motion  of  the  system  will  not  recur  periodically  through  the  same 
series  of  configurations.  There  are  in  general,  however,  i  distinct 
determinate  displacements,  which  we  shall  call  the  normal  displace- 
mefits,  fulfilling  the  condition,  that  if  any  one  of  them  be  produced 
alone,  and  the  system  then  left  to  itself  for  an  instant  at  rest,  this 
displacement  will  diminish  and  increase  periodically  according  to 
a  simple  harmonic  function  of  the  time,  and  consequently  every 
particle  of  the  system  will  execute  a  simple  harmonic  movement  in 
the  same  period.  This  result,  we  shall  see  later,  includes  cases  in 
which  therQ  are  ..n  infinite  number  of  degrees  of  freedom;  as,  for 
instance,  a  stretched  cord;  a  mass  of  air  in  a  closed  vessel;  waves 
in  water,  or  oscillations  of  water  in  a  vessel  of  limited  extent,  or  an 
elastic  solid;  and  in  these  applications  it  gives  the  theory  of  the 
so<alIed  'fundamental  vibration,'  and  successive  'harmonics*  of  the 
cord,  and  of  all  the  different  possible  simple  modes  of  vibration  in 
the  other  cases.  In  all  these  cases  it  is  convenient  to  give  the  name 
'fundamental  mode'  to  any  one  of  the  possible  simple  harmonic 
vibrations,  and  not  to  restrict  it  to  the  gravest  simple  harmonic  mode, 
as  has  been  hitherto  usual  in  respect  to  vtbrating  cords  and  organ- 
pipes. 

The  whole  kinetic  energy  of  any  complex  motion  of  the  system  is 
equal  to  the  sum  of  the  kinetic  energies  of  the  fundamental  constitu- 
ents; and  the  potential  energy  of  any  displacements  is  equal  to  the  sum 
of  the  potential  energies  of  its  normal  components.  Corresponding 
theorems  of  normal  constituents  and  fundamental  modes  of  motion, 
and  the  summation  of  their  kinetic  and  potential  energies  in  complex 
motions  and  displacements,  hold  for  motion  in  the  neighbourhood  of 
a  configuration  of  mrstable  equilibrium.  In  this  case,  some  or  all  ol 
the  constituent  motions  are  fallings  away  from  the  position  of  equi- 
librium (according  as  the' potential  energies  of  the  constituent  normal 
vibrations  are  negative). 

291.  If,  as  may  be  in  particular  cases,  the  periods  of  the  vibrations 
for  two  or  more  of  the  nojnfial  displacements  are  equal,  any  displace- 
ment compounded  of  thctn'will  also  fulfil  the  condition  of  a  normal 
displacement.  And  if  the  system  be  displaced  according  to  any  one 
such  normal  displacement,  and  projected  with  velocity  corresponding 
to  another,  it  will  execute  a  movement,  the  resultant  of  two  simple 
harmonic  movements  in  equal  periods.     The  graphic  representation 


DYNAMICAL  LAWS  AND  FRIXCIFLES. 


99 


of  the  variation  of  the  corresponding  co-ordinates  of  the  system,  laid 
down  as  two  rectangular  co-ordinates  in  a  [ilaiie  dl.igiam,  will  con- 
se(]Liently  (.§  S2)  be  a  circle  or  an  cllij)5c;  v.'r.ich  v;iil  therefore,  of 
course,  be  the  form  of  the  crbit  of  any  particle  of  the  system  whicli 
has  a  distinct  direction  of  motion,  for  two  of  the  displacements  in 
question.  But  it  must  be  remembered  that  some  cf  the  principal 
pa.rts  may  have  only  one  degree  of  freedom;  or  even  that  each  \y\x\. 
of  the  system  may  have  only  one  degree  of  freedom  (as,  for  instance, 
if  the  s}-stem  is  composed  of  a  set  of  particles  each  constrained  to 
remain  l^w  a  given  line,  or  of  rigid  bodies  on  fixed  axes,  mutually 
int'.uencing  one  another  by  elastic  cords  or  otherwise).  In  such  a 
case  as  the  last,  no  i>article  of  the  system  can  move  otherwise  than 
in  oiie  line,  and  the  ellipse,  circle,  or  other  graphical  representation 
of  the  composition  of  the  harmonic  motions  of  the  system,  is  merely 
an  aid  to  comprehension,  and  not  a  representation  of  any  mo'iion 
actually  taking  place  m  any  part  of  the  svstem. 

292.  In  nature,  as  has  been  said  above  (§  250;,  every  system 
uninthienced  by  matter  external  to  it  is  conservative,  v.-hen  the 
ultimate  molecular  motions  constituting  heat,  light,  and  magnetism, 
and  the  potential  energy  of  chemical  afiinities,  are  taken  into  account 
along  with  the  [xalpable  motions  and  measurable  forces.  But  (§  247) 
practically  we  are  obliged  to  admit  forces  of  friction,  and  resistances 
of  the  other  classes  there  enumerated,  as  causing  losses  of  energy  to 
be  reckoned,  in  abstract  dynamics,  without  regard  to  the  equivalents 
of  heat  or  other  molecular  actions  which  they  generate.  Hence  wlien 
such  resistances  are  to  be  taken  into  account,  forces  opi-io-ed  to  the 
motions  of  various  parts  of  a  systein  must  be  introduced  into  the 
equations.  According  to  the  approximate  knowledge  which  we  have 
from  experiment,  these  forces  are  independent  of  the  velocities  when 
due  to  the  friction  of  solids^  and  are  siinjily  jjroportional  to  the 
velocities  when  due  to  tluid  viscosity  directly,  or  to  electric  or;nagnetic 
influences,  with  corrections  de])ending  on  varying  temperature,  and 
on  the  varying  configuration  of  the  system.  In  consequence  of  the 
last-mentioned  cause,  the  resistance  of  a  real  liquid  (which  is  always 
more  or  less  viscous)  against  a  body  moving  very  rapidly  through  it, 
and  leaving  a  great  deal  of  irregular  motion,  such  as  'eddies,'  in  its 
wake,  seems  to  be  nearly  in  proportion  to  the  square  of  the  velocity;, 
although,  as  Stokes  has  shown,  at  the  lowest  speeds  the  resistance 
is  probably  in  simple  proportion  to  the  velocity,  and  for  all  speeds 
may,  it  is  probable,  be  approximately  expressed  as  the  sum  of  two 
terms,  one  simply  as  the  velocity,  and  the  other  as  the  square  of  the 
velocity  If  a  solid  is  started  from  rest  in  an  incompressible  fluid, 
the  mitial  law  of  resistance  is  no  doubt  simple  proportionality  to  the 
velocity,  (however  great,  if  suddenly  enough  given;)  until  by  the 
gradual  growth  of  eddies  the  resistance  is  increased  gradually  till  it 
comes  to  fulfil  Stokes's  law. 

293.  The  effect  of  friction  of  solids  rubbing  one  against  another 
is  simply  to  render  impossible  the  infinitely  small  vibrations  with  which 


loo  PRELIMINARY. 

'we  are  now  particularly  concerned ;  and  to  allow  any  system  m  which 
it  is  present,  to  rest  balanced  when  displaced  within  certain  finite 
limits,  from  a  configuration  of  fi"ictionless  equilibrium.  In  mechanics 
it  is  easy  to  estimate  its  effects  with  sufficient  accuracy  when  any 
practical  case  of  finite  oscillations  is  in  question.  But  the  other 
classes  of  dissipative  agencies  give  rise  to  resistances  simply  as  the 
velocities,  without  the  corrections  referred  to,  when  the  motions  are 
infinitely  small,  and  can  never  balance  the  system  in  a  configuration 
deviating  to  any  extent,  however  small,  from  a  configuration  of 
equilibrium  without  friction.  In  the  theory  of  infinitely  small  vibra- 
tions, they  are  to  be  taken  into  account  by  adding  to  the  expressions 
for  the  generalized  components  of  force,  terms  consisting  of  the 
generalized  velocities  each  multiplied  by  a  constant,  winch  gives  us 
equations  still  remarkably  amenable  to  rigorous  mathematical  treat- 
ment. The  result  of  the  integration  for  the  case  of  a  single  degree 
of  freedom  is  very  simple;  and  it  is  of  extreme  importance,  both  for 
the  explanation  of  many  natural  phenomena,  and  for  use  in  a  large 
variety  of  experimental  investigations  in  Natural  Philosophy.  Partial 
conclusions  from  it,  in  the  first  place,  stated  in  general  terms,  are 
as  follows: — 

294.  If  the  resistance  per  unit  velocity  is  less  than  a  certain  limit, 
in  any  particular  case,  the  motion  is  a  simple  harmonic  oscillation, 
with  amplitude  decreasing  by  equal  proportions  in  equal  successive 
intervals  of  time.  But  if  the  resistance  exceeds  this  limit,  the  system, 
when  displaced  from  its  position  of  equilibrium  and  left  to  itself, 
returns  gradually  towards  its  position  of  equilibrium,  never  oscillating 
through  it  to  the  other  side,  and  only  reaching  it  after  an  infinite 
time. 

In  the  unresisted  motion,  let  if  be  the  rate  of  acceleration,  when 

the  displacement  is  unity;  so  that  (§  74)  we  have  T-  — :  and  let  the 

rate  of  retardation  due  to  the  resistance  corresponding  to  unit  velocity 
be  k.  Then  the  motion  is  of  the  oscillatory  or  non-oscillatory  class 
according  as  I^  <{27if  or  k^-^  (2;?)^     In  the  first  case,  the  period  of 

the  oscillation  is  increased,  by  tfie  resistance,  from  T\.o  T  j-^ — r7?rr- 

.  •'  («  -i^)*' 

and  the  rate  at  which  the  Napierian  logarithm  of  the  amplitude 
diminishes  per  unit  of  time  is  \k. 

295.  An  indirect  but  very  simple  proof  of  this  important  propo- 
sition may  be  obtained  by  means  of  elementary  mathematics  as 
follows  ■ — A  point  describes  a  logarithmic  spiral  with  uniform  angular 
velocity  about  the  pole — find  the  acceleration. 

Since  the  angular  velocity  of  SP  and  the  inclination  of  this  line 
to  the  tangent  are  each  constant,  the  linear  velocity  of  P  is  as  SP. 
Take  a  length  PT,  equal  to  n  SP,  to  represent  it.  Then  the 
hodograph,  the  locus  of  /,  where  Sp  is  parallel  and  equal  to  PT,  is 
evidently  another  logarithmic  spiral  similar  to  the  former,  and  de- 
scribed with  the  same  uniform  angular  velocity.     Hence  (^  35,  49) 


DYXAMICAL  LAWS  AND  FRi;\CIPLES.  loi 

//,  the  acceleration  required,  is  equPil  to  ;/  Sp,  a.'-id  maket  with  Sp  an 
angle  Spt  equal  to  SPT.     Hence,  if  Pu  be  drawn  parallel  and  equal 


to//,  and  uv  parallel  to  PT,  the  whole  acceleration//  or  Pn  may  be 
resolved  into  Pv  and  vu.  Now  Pvu  is  an  isosceles  triangle,  whose 
base  angles  {v,  u)  are  each  equal  to  the  constant  angle  of  the  spiral. 
Hence  Pv  and  vu  bear  constant  ratios  to  Pu,  and  therefore  to  SP 
and  /'/'respectively. 

The  acceleration,  therefore,  is  composed  of  a  central  attractive 
part  proportional  to  the  distance,  and  a  tangential  retarding  part 
proportional  to  the  velocity. 

And,  if  the  resolved  part  of  /"s  motion  parallel  to  any  line  in  the 
plane  of  the  spiral  be  considered,  it  is  obvious  that  m  it  also  the 
acceleration  will  consist  of  two  parts — one  directed  towards  a  point 
in  the  line  (the  projection  of  the  pole  of  the  spiral),  and  proportional 
to  the  distance  from  it ;  the  other  proportional  to  the  velocity,  but 
retarding  the  motion. 

Hence  a  particle  which,  unresisted,  would  have  a  simple  harmonic 
motion,  has,  when  subject  to  resistance  proportional  to  its  velocity, 
a  motion  represented  by  the  resolved  part  of  the  spiral  motion  just 
described. 

296.  If  a  be  the  constant  angle  of  the  spiral,  w  the  angular  velocity- 
of  SP,  we  have  evidently 

PT.  sin  a  =  SP.  w      But  PT=  nSP,  so  that  n  =  '-^  . 

sin  a 

Hence  Pv  =  Pu=p/ =  nSp  =  nPT=  n\  SP 

and  vu  -  2Pv  .  cos  a  =  2«  cos  aPT=  k  .  /'/'(suppose). 

Thus  the  central  force  at  unit  distance  is  ;;'=   .   ,     ,   and  the  co- 

sin  a 

^    .  .         .  .      ,  2ajC0Sa 

emcient  of  resistance  is  /e  =  2n  cos  a  =  — ■. . 

sin  a 

The  time  of  oscillation  in  the  resolved  motion  is  evidently  — ;  but, 

if  there  had  been  no  resistance,  the  properties  of  simple  harmonic 


UNIVEKSITY  (M-'  c^^  frn:;xiA 
SANTA   BAIIUAICA 


102  PRELIMINARY. 

motion  show  that  it  would  have  been  —  :  so  that  it  is  increased  by 

n  •' 

/ J2 

the  resistance  in  the  ratio  coseca  to  i.  or  ;z  to  *  /  ;^'--  . 

V  4 

The  rate  of  diminution  of  SP  is  evidently 

PT .  CQSa-n  COS  a  SP=^-  SP; 

2 

that  is,  SP  diminishes  in  geometrical  progression  as  time  increases, 
the  rate  being  -  per  unit  of  time  per  unit  of  length.  By  an  ordinary 
result  of  arithmetic  (compound  interest  payable  every  instant)  the 
diminution  of  log.  SP  in  unit  of  time  is  -. 

2 

This  process  of  solution  is  only  applicable  to  resisted  harmonic 

k  k 

vibrations  when  n  is  greater  than  -.     When  7i  is  not  greater  than  - 

the  auxiliary  curve  can  no  longer  be  a  logarithmic  spiral,  for  the 
moving  particle  never  describes  more  than  a  finite  angle  about  the 
pole.  A  curve,  derived  from  an  equilateral  hyperbola,  by  a  process 
somewhat  resembling  that  by  which  the  logarithmic  spiral  is  deduced 
from  a  circle,  may  be  introduced ;  but  then  the  geometrical  method 
ceases  to  be  simpler  than  the  analytical  one,  so  that  it  is  useless  to 
pursue  the  investigation  farther,  at  least  from  this  point  of  view. 

297.  The  general  solution  of  the  problem,  to  find  the  motion  of 
a  system  having  any  number,  z,  of  degrees  of  freedom,  when  infinitely 
little  disturbed  from  a  position  oi  equilibrium,  and  left  to  move  subject 
to  resistanc«3  proportional  to  velocities,  shows  that  the  whole  motioA 
may  be  resolved,  in  general  determinately,  into  2/  different  motions 
each  either  simple  harmonic  wdth  amplitude  diminishing  according  to 
the  law  stated  above  (§  294),  or  non-oscillatory,  and  consisting  of 
equi-proportionate  diminutions  of  the  components  of  displacement 
in  equal  successive  intervals  of  time. 

298.  When  the  forces  of  a  system  depending  on  configuration, 
and  not  on  motion,  or,  as  we  may  call  them  for  brevity,  the  forces 
of  position,  vioL-ite  the  law  of  conservatism,  we  have  seen  (§  244) 
that  energy  without  limit  may  be  drawn  from  it  by  guiding  it  per- 
petually through  a  returning  cycle  of  configurations,  and  we  have 
inferred  that  in  every  real  system,  not  supplied  with  energy  from 
without,  the  forces  of  position  fulfil  the  conservative  law.  But  it  is 
easy  to  arrange  a  system  artificially,  in  connexion  with  a  source  of 
energy,  so  that  its  forces  of  position  shall  be  non-conservative ;  and 
the  consideration  of  the  kinetic  eftects  of  such  an  arrangement, 
especially  of  its  oscillations  about  or  motions  round  a  configuration 
of  equilibrium,  is  most  instructive,  by  the  contrasts  which  it  presents 

'to  the  phenomena  of  a  natural  system. 

299.  But  although,  when  the  equilibrium  is  stable,  no  possible 


DYNAMICAL  LAWS  AND  PRINCIPLES.  103 

infinitely  small  displacement  and  velocity  given  to  the  system  can 
cause  it,  when  left  to  itself,  to  go  on  moving  either  farther  and  farther 
away  till  a  finite  displacement  is  reached,  or  till  a  finite  velocity  is 
acquired;  it  is  very  remarkable  that  stability  should  be  possible,' 
considering  that  even  in  the  case  of  stability  an  endless  increase  of 
velocity  may,  as  is  easily  seen  from  §  244,  be  obtained  merely  by- 
constraining  the  system  to  a  particular  closed  course,  or  circuit  of 
configurations,  nowhere  deviating  by  more  than  an  infinitely  small 
amount  from  the  configuration  of  equilibrium,  and  leaving  it  at 
rest  anywhere  in  a  certain  part  of  this  circuit.  This  result,  and 
the  distinct  peculiarities  of  the  <:ases  of  stability  and  instability,  are 
sufiiciently  illustrated  by  the  simplest  possible  example, — that  of  a 
material  particle  rrioving  in  a  plane. 

300.  There  is  scarcely  any  question  in  dynamics  more  important 
for  Natural  Philosophy  than  the  stability  or  instability  of  motion.  We 
therefore,  before  concluding  this  chapter,  propose  to  give  some 
general  explanations  and  leading  principles  regarding  it. 

A  'conservative  disturbance  of  motion'  is  a  distiirbance  in  the 
motion  or  configuration  of  a  conservative  system,  not  altering  the 
sum  of  the  potential  and  kinetic  energies.  A  conservative  disturb- 
ance of  the  /notion  through  any  particular  configuration  is  a  change 
in  velocities,  or  component  velocities,  not  altering  the  whole  kinetic 
energy.  Thus,  for  example,  a  sonservative  disturbance  of  the  motion 
of  a  particle  through  any  point,  is  a  change  in  the  direction  of  its 
motion,  unaccompanied  by  change  of  speed. 

301.  The  actual  motion  of  a  system,  from  any  particular  con- 
figuration, is  said  to  be  stable  if  every  possible  infinitely  small  con- 
servative disturbance  of  its  motion  through  that  configuration  may 
be  compounded  of  conservative  disturbances,  any  one  of  which  would 
give  rise  to  an  alteration  of  motion  which  would  brin^  the  system 
again  to  some  configuration  belonging  to  the  undisturbed  path,  in 
a  finite  time,  and  without  more  than  an  infinitely  small  digression. 
If  this  condition  is  not  fulfilled,  the  motion  is  said  to  be  unstable. 

302.  For  example,  if  a  body,  A,  be  supported  on  a  fixed  vertical 
axis ;  if  a  second,  B,  be  supported  on  a  parallel  axis  belonging  to 
the  first;" a  third,  C,  similarly  supported  on  B,  and  so  on;  and  if 
B,  C,  etc.,  be  so  placed  as  to  have  each  its  centre  of  inertia'as  far  as 
\possible  from  the  fixed  axis,  and  the  whole  set  in  motion  with 
a  common  angular  velocity  about  this  axis,  the  motion  will  be 
thoroughly  stable.  If,  for  instance,  each  of  the  bodies  is  a  flat 
rectangular  board  hinged  on  one  edge,  it  is  obvious  that  the  whole 
system  will  be  kept  stable  by  centrifugal  force,  when  all  are  in  one 
plane  and  as  far  out  from  the  axis  as  possible.  But  if  A  consists 
partly  of  a  shaft  and  crank,  as  a  common  spinning-wheel,  or  the  fly- 
wheel and  crank  of  a  steam-engine,  and  if  B  be  supported  on  the 
crank-pin  as  axis,  and  turned  inwards  (towards  the  fixed  axis,  or 
across  the  fixed  axis),  then,  even  although  the  centres  of  inertia  of  Q 


104  PRELIMINARY. 

D,  etc,  are  placed  as  far  from  the  fixed  axis  as  possible,  consistent 
with  this  position  of  B,  the  motion  of  the  system  will  be  unstable. . 

803.  The  rectilinear  motion  of  an  elcTngated  body  lengthwise,  or 
of  a  flat  disc  edgewise,  through  a  fluid  is  unstable.  But  the  motion  of 
either  body,  with  its  length  or  its  broadside  perpendicular  to  the 
direction  of  motion,  is  stable.  Observation  proves  the  assertion  we 
have  just  made,  for  real  fluids,  air  and  water,  and  for  a  great  variety 
of  circumstances  affecting  the  motion ;  and  we  shall  return  to  the 
subject  later,  as  being  not  only  of  great  practical  importance,  but 
profoundly  interesting,  and  by  no  means  difficult  in  theory. 

304.  The  motion  of  a  single  particle  affords  simpler  and  not  less 
instructive  illustrations  of  stability  and  instability.  Thus  if  a  weight, 
hung  from  a  fixed  point  by  a  light  inextensible  cord,  be  set  in  motion 
so  as  to  describe  a  circle  about  a  vertical  line  through  its  position  of 
equilibrium,  its  motion  Is  stable.  For,  as  we  shall  see  later,  if  dis- 
turbed infinitely  little  in  direction'  without  gain  or  loss  of  energy,  it 
will  describe  a  sinuous  path,  cutting  the  undisturbed  circle  at  points 
successively  distant  from  one  another  by  definite  fractions  of  the 
circumference,  depending  upon  the  angle  of  inclination  of  the  string 
to  the  vertical.  When  this  angle  is  very  small,  the  motion  is  sensibly 
the  same  as  that  of  a  particle  confined  to  one  plane  moving  under 
the  influence  of  an  attractive  force  towards  a  fixed  point,  simply  pro- 
portional to  the  distance ;  and  the  disturbed  path  cuts  the  undisturbed 
circle  four  times  in  a  revolution.  Or  if  a  particle  confined  to  one 
plane,  move  under  the  influence  of  a  centre  in  this  plane,  attracting 
with  a  force  inversely  as  the  square  of  the  distance,  a  path  infinitely 
little  disturbed  from  a  circle  will  cut  the  circle  twice  in  a  revolution. 
Or  if  the  law  of  central  force  be  the  n\h  power  of  the  distance,  and  if 
«  -t-  3  be  positive,  the  disturb'ed  path  will  cut  the  undisturbed  circular 

orbit  at  successive  angular  intervals,"  each  equal  to  -  .     But  the 

motion  will  be  unstable  if  «  be  negative,  and-«>  3. 

305.  The  case  of  a  particle  moving  on  a  smooth  fixed  surface 
under  the  influence  of  no  other  force  than  that  of  the  constraint,  and 
therefore  always  moving  along  a  geodetic  line  of  the  surface,  afi"ords 
extremely  simple  illustrations  of  stability  and  instabihty.  For  instance, 
a  pardcle  placed  on  the  inner  circle  of  the  surface  of  an  anchor-ring, 
and  projected  in  the  plane  of  the  ring,  would  move  perpetually  in  that 
circle,  but  unstably,  as  the  smallest  disturbance  would  clearly  send  it 
away  from  this  path,  never  to  return  until  after  a  digression  round  the 
outer  edge.  (We  suppose  of  course  that  the  particle  is  held  to  the 
surface,  as  if  h  were  placed  in  the  infinitely  narrow  space  between  a 
solid  ring  and  a  hollow  one  enclosing  it.)  But  if  a  particle  is  placed 
on  the  outermost,  or  greatest,  circle  of  the  ring,  and  projected  in  its 
plane^.an  infinitely  small  disturbance  will  cause  it  to  describe  a  sinuous 
path  cutting  the  circle  at  points  round  it  successively  distant  by  angles 


DYNAMICAL   LAWS  AXD  PRLXCLPLES.         105 

each  equal  to  tt     ./-  ,  and  therefore  at  intervals  of  time,  each  equal  to 

-  /  - ,  where  a  denotes  the  radius  of  that  circle,  w  the  angular  velocity 
ui\'  a  '  =>  } 

in  it,  and  ^  the  radius  of  the  circular  cross  section  of  the  ring.  This 
is  proved  by  remarking  that  an  infinitely  narrow  band  from  the  outer- 
most part  of  the  ring  has,  at  each  point,  a  and  d  from  its  principil 
radii  of  curvature,  and  therefore  (§  134)  has  for  its  geodetic  lines  the 
great  circles  of  a  sphere  of  radius  Jal),  upon  which  it  may  be  bent. 

306.  In  all  these  cases  the  undisturbed  motion  has  been  circular 
or  rectilineal,  and,  when  the  motion  has  been  stable,  the  effect  of  a 
disturbance  has  been  periodic,  or  recurring  with  the  same  phases  in 
equal  successive  intervals  of  time.  An  illustration  of  thoroughly  stable 
motion  in  which  the  effect  of  a  disturbance  is  not  'periodic,'  is  pre- 
sented by  a  particle  sliding  down  an  inclined  groove  under  the  action 
of  gravity.  To  take  the  simplest  case,  we  may  consider  a  particle 
sliding  down  along  the  lowest  straight  line  of  an  inclined  hollow 
cylinder.  If  slightly  disturbed  from  this  straight  line,  it  will  oscillate 
on  each  side  of  it  perpetually  in  its  descent,  but  not  with  a  uniform 
periodic  motion,  though  the  durations  of  its  excursions  to  each  side  of 
the  straight  line  are  all  equal. 

307.  A  very  curious  case  of  stable  motion  is  presented  by  a  particle 
constrained  to  remain  on  the  surface  of  an  anchor-ring  fixed  in  a 
vertical  plane,  and  projected  along  the  great  circle  from  any  point  of 
it,  with  any  velocity.  An  infinitely  small  disturbance  will  give  rise  to 
a  disturbed  motion  of  which  the  path  will  cut  the  vertical  circle  over 
and  over  again  for  ever,  at  unequal  intervals  of  time,  and  unequal 
angles  of  the  circle ;  and  obviously  not  recurring  periodically  in  any 
cycle,  except  with  definite  particular  values  for  the  whole  energy,  some 
of  which  are  less  and  an  infinite  number  are  greater  than  that  which 
just  suffices  to  bring  the  particle  to  the  highest  point  of  the  ring.  The 
full  mathematical  investigation  of  these  circumstances  would  afford  an 
excellent  exercise  in  the  theory  of  differential  equations,  but  it  is  not 
necessary  foi:  oijr  present  illustrations. 

308.  In  this  case,  as  in  all  of  stable  motion  with  only  two  degrees 
of  freedom,  which  we  have  just  considered,  there  has  been  stability 
throughout  the  motion  ;  and  an  infinitely  small  disturbance  from  any 
point  of  the  motion  has  given  a  disturbed  path  which  intersects  the 
undisturbed  path  over  and  over  again  at  firtite  int^rals  of  time. 
But,  for  the  sake  of  simplicity,  at  present  confining  our  attention  to 
two  degrees  of  freedom,  we  have  a  limited  stability  in  the  raotion  of  an 
unresisted  projectile,  which  satisfies  the  criterion  of  stability  only  at 
points  of  its  upward,  not  of  its  downward,  path.  Thus  if  MOFQ  be 
the  path  of  a  projectile,  and  if  at  O  it  be  disturbed  by  an  infinitely 
small  force  either  way  perpendicular  to  its  instantaneous  direction  of 
motion,  the  disturbed  path  will  cut  the  undisturbed  infinitely  near 
the  point  /"where  the  direction  of  motion  is  perpendicular  to  that  at  0' 


I  c6  PRELIMINAR  Y. 

as  ^ve  e"S'"h'  see  by  considering  that  the  hne  joining  two  particles  pro-"^ 
jected  Irorn  one  point  at  the  same  instanc  with  equal  velocities  in  the 


\\- 


directions  of  any  two  lines,  will  always  remain  perpendicular  to  the 
line  bisecting  the  angle  between  these  two. 

309.  The  principle  of  varying  action  gives  a  mathematical  criterion 
for  stability  or  instability  in  every  case  of  motion.  Thus  in  the  first 
place  it  is  obvious  (§§  308,  3x1),  that  if  the  action  is  a  true  minimum 
in  the  motion  of  a  system  from  any  one  configuration  to  the  con- 
figuration reached  at  any  other  tame,  however  much  later,  the  motion 
is  thoroughly  unstable.  For  instance,  in  the  motion  of  a  particle  con- 
strained to  remain  on  a  smooth  fixed  surface,  and  uniniiuenced  by 
gravity,  the  action  is  simply  the  length  of  the  path,  multiplied  by  the 
constant  velocity.  Hence  in  the  particular  case  of  a  particle  unin- 
fluenced by  gravity,  moving  round  the  inner  circle  in  the  plane  of  an 
anchor-ring  considered  above,  the  action,  or  length  of  path,  is  clearly 
a  minimum  for  any  one  point  to  the  point  reached  at  any  subsequent 
time.  (The  action  is  not  merely  a  minimum,  but  is  the  least  possible, 
from  any  point  of  the  circular  path  to  any  other,  through  less  than  half 
a  circumference  of  the  circle.)  On  the  other  hand,  although  the  path 
from  any  point  in  the  greatest  circle  of  the  ring  to  any  other  at  a  dis- 
tance from  it  along  the  circle,  less  than  tt  Jab,  is  clearly  least  possible 
if  along  the  circumference  ;  the  path  of  absolutely  least  length  is  not 
along  the  circumference  between  two  points  at  a  greater  circular 
distance  than  -n-Jab  from  one  another,  nor  is  the  path  along  the 
circumference  between  them  a  minimum  at  all  in  this  latter  case.  On 
any  surface  whatever  which  is  everywhere  anticlastic,  or  along  a  geo- 
detic of  any  surface  which  passes  altogether  through  an  anticlastic 
region,  the  motion  is  thoroughly  unstable.  For  if  it  were  stable  from 
any  point  O,  we  should  have  the  given  undisturbed  path,  and  the 
disturbed  path  from  O  cutting  it  at  some  point  Q — two  different 


DYNAMICAL  LAWS  AND  PRINCIPLES.        107 

geo9et;c  lines  joining  two  points ;  which  is  impossible  on  any 
anticlastic  surface,  inasmuch  as  the  sum  of  the  exterior  angles  of 
any  closed  figure  of  geodetic  lines  exceeds  four  right  angles  when 
the  integral  curvature  of  the  enclosed  area  is  negative,  which 
is  the  case  for  every  portion  of  surface  thoroughly  anticlastic. 
But,  on  the  other  hand,  it  is  easily  proved  that  if  we  have  an  endless 
rigid  band  of  curved  surface  everywhere  synclastic,  ■with  a  geodetic 
line  running  through  its  middle,  the  motion  of  a  particle  projected 
along  this  line  will  be  stable  throughout,  and  an  infinitely  slight  disturb- 
ance will  give  a  disturbed  path  cutting  the  given  undisturbed  path  again 
and  again  for  ever  at  successive  distances  differing  according  to  the 
different  specific  curvatures  of  the  intermediate  portions  of  the  surface. 

310.  If,  from  any  one  configuration,  two  courses  diflfering  infinitely 
little  from  one  another,  have  again  a  configuration  in  common,  this 
second  configuration  will  be  called  a  kinetic  focus  relatively  to  the 
first:  or  (because  of  the  reversibility  of  the  motion)  these  two  con- 
figurations will  be  called  conjugate  kinetic  foci.  ,Optic  foci,  if  for 
a  r.oment  we  adopt  the  corpuscular  theory  of  light,  are  included  as 
a  particular  case  of  kinetic  foci  in  general.  But  it  is  not  difficult 
to  prove  that  there  must  be  finite  intervals  of  space  and  time  be- 
tween two  conjugate  foci  in  every  motion  of  ever)-  kind  of  system, 
only  provided  the  kinetic  energ-y  does  not  vpjiish. 

311.  Now  it  is  obvious  that,  provided  only  a  sufficiently  short 
course  is  considered,  the  action,  in  any  natural  motion  of  a  system, 
is  less  than  for  any  other  course  between  its  terminal  configurations. 
It  will  be  proved  presently  (§  314)  that  the  first  configuration  up  to 
which  the  action,  reckoned  from  a  given  initial  configuration,  ceases 
to  be  a  minimum,  is  the  first  kinetic  focus;  and  conversely,  that  when 
the  first  kinetic  focus  is  passed,  the  action,  reckoned  from  the  initial 
configuration,  ceases  to  be  a  minimum ;  and  therefore  of  course  can 
never  again  be  a  minimum,  because  a  course  of  shorter  action, 
deviating  infinitely  little  from  it,  can  be  found  for  a  part,  without 
altering  the  remainder  of  the  whole,  natural  course. 

'312.  In  such  statements  as  this  it  will  frequently  te  convenient 
to  indicate  particular  configurations  of  the  system  by  single  letters, 
as  0,  P,  (2,  i? ;  and  any  particular  course,  in  which  it  moves  through 
configurations  thus  indicated,  will  be  called  the  course  O...P...Q...P. 
The  adiofi  in  any  natural  course  will  be  denoted  simply  by  the 
terminal  letters,  taken  in  the  order  of  the  motion.  Thus  OR  will 
denote  the  action  from  O  io  R ;  and  therefore  0R  =  -  RO.  When 
there  are  more  real  natural  courses  from  O  to  R  than  one,  the 
analytical  expression  for  OR  will  have  more  than  one  real  value; 
and  it  may  be  necessary  to  specify  for  which  of  these  courses  the 
action  is  reckoned.     Thus  we  mav  have 

OR  for  'O...E...R, 

OR[oxO...E...R, 

ORioTO...E"...R, 
three  different  values  of  one  algebraic  irrational  expression. 


io8 


I'RELTMINARY. 


313.  In  terms  of  this  notation  the  preceding  statement  (§311) 
tnay  be  expressed  thus  : — If,  for  a  conservative  system,  moving  on 
a  certain  course  0...P...0' ...P\  the  first  kinetic  focus  conjugate 
to  C  be  0' ,  the  action  OP,  in  this  course,  will  be  less  than  the  action 
along  any  other  course  deviating  infinitely  little  from  it:  but,  on  the 
other  hand,  OP'  is  greater  than  the  actions  in  some  courses  from 
O  to  P'  deviating  infinitely  little  from  the  specified  natural  course 

o...p...b'...p'. 

314.  It  must,  not  be  supposed  that  the  action  along  OP  is  neces- 
sarily ihe  least  possible  from  O  to  P.  There  are,  in  fact,  cases  in 
which  the  action  ceases  to  be  least  of  all  possible,  before  a  kinetic 
focus  is  reached.  Thus  if  OEAPO'E'A'  be  a  sinuous  geodetic  line 
cutting  the  outer  circle  of  an  anchor-ring,  or  the  equator  of  an  oblate 
spheroid,  in  successive  points  6',  ^,  ^',  it  is  easily  seen  that  O',  the 
first  kinetic  focus  conjugate  to  O,  must  he  somewhat  beyond  A. 
But  the  length  OEAP,  although  a  minimum  (a  stable  position  for 
a  stretched  string),  is  not  the  shortest  distance  on  the  surface  from 
O  to  P,  as  this  must  obviously  be  a  line  lying  entirely  on  one  side 


of  the  great  circle.  From  O  to  any  point,  Q,  short  of  ^,  the  distance 
along  the  geodetic  OEQA  is  clearly  the  least  possible :  but  if  Q  be 
near  enough  to  A  (that  is  to  say,  between  A  and  the  point  in  which 
the  envelop  of  the  geodetics  drawn  from  O,  cuts-  OEA),  there  will 
also  be  two  other  geodetics  from  O  to  Q.  The  length  of  one  of 
these  will  be  a  minimum,  and  that  of  the  other  not  a  minimum.* 
If  Q  is  ipoved  forward  to  A,  the  former  becomes  OE,A,  equal  and 
similar  to  OEA,  but  on  the  other  side  of  the  great  circle :  and  the 
latter  becomes  the  great  circle  from  O  to  A.  If  now  Q  be  moved 
on  to  P,  beyond  A,  the  minimum  geodetic  OEAP  ceases  to  be  the 
less  of  the  two  minima,  and  the  geodetic  OEPlying  altogether  on  the 
other  side  of  the  great  circle  becomes  the  least  possible  line  from 
O  to  P.  But  until  P  is  advanced  beyond  the  point  O',  in  which  it 
is  cut  by  another  geodetic  from  O  lying  infinitely  nearly  along  it, 
the  length  OEAP  remains  a  minimum  .according  to  the  general 
proposition  of  §  31 1. 


DYNAMICAL   LAWS  AND   PRIXCIPLES.         109 

316.  As  it  has  been  proved  that  the  action  from  any  configuration 
ceases  to  be  a  minimum  at  the  first  conju^'aie  kinciic  focus,  we  see 
immc  'iatcly  that  if  (9' be  the  first  kinetic  focus  conJu_;.ue  to  O,  reached 
after  passing  <9,  no  two  configurations  on  this  coarse  from  O  to  O' 
can  be  kinetic  foci  to  one  another.  For,  the  actiun  from  O  ju-t 
ct...sing  to  be  a  minimum  when  O'  is  reached,  the  action  b.j".'.',ccn  any 
two  intermediate  configurations  of  tlie  same  course  is  necessarily  a. 
minimum. 

316.  When  there  are  I  degrees  of  freedom  to  move  tl;erc^~are  in 
general,  on  any  natural  course  from  any  particular  configuratiorH  O, 
at  least  /-  1  kinetic  foci  conjugate  to  O.  Thus,  for  example,  on  the 
course  of  a  ray  of  light  em.anating  from  a  lunrinous  point  C,  and  pass- 
ing through  the  centre  of  a  convex  lens  held  obliquely  to  its  path, 
there  are  two  kinetic  foci  conjugate  to  O,  as  defined  abo\'e,  being  the 
points  in  which  the  line  of  the  central  ray  is  cut  by  the  so-called 
'focal  lines"  of  a  pencil  of  rays  diverging  from  O  and  made  con- 
vergent after  passing  through  th9  lens.  But-  some  or  all  of  these 
kinL-tic  foci  may  be  on  the  cour5(J'-previous  to  O ;  as,  for  instance,  in 
the  case  of  a  common  projectile  wn.en  its  course  passes  obliquely 
downw.irds  through  O.  Or  some  or  all  may  be  lost,  as  when, 
in  the  optical  illustration  just  referred  to.  the  L-ns  is  pnly  strong 
enough  to  produce  convergence  in  one  of  the  ])rincipal  planes,  or 
too  weak  to  produce  convergence  in  either.  Thus  also  in  the 
case  of  the  undisturbed  rectilineal  motion  of  a  point,  or  in  the 
motion  of  a  point  uniiifluenced  by  force,  on  an  anticlastic  surface 
(§  309)  there  are  no  real  kinetic  foci.  In  the  motioti  of  a  pro- 
jectile (not  confined  to  one  vertical  plane)  there  can  be  only  one 
kinetic  focus  on  each  'path,  conjugate  to  one  given  point ;  though 
there  are  three  degrees  of  freedom.  Again,  there  may  be  any  number 
more  tkian  i—i  of  foci  in  one  course,  all  conjugate  to  one  con- 
figuration, as  ior  instance  on  the  course  of  a  particle,  uniruluenced  by 
force,  moving  roimd  the  surfice  of  an  anchor-ring,  along  either  the 
outer  great  circle,  or  along  a  sinuous  geodetic  such  as  v,-e  have  con- 
sidered in  .§  311,  in  which  clearly  there  are  aw  infinite  number  of  foci 
each  conjugate  to  any  one  point  of  the  path,  at  equal  succe-^sive  dis- 
tances from  one  another. 

317.  If/—  I  distinct"  courses  from  a  configuration  C,  each  dilTering 

infinitely  little  from  a  certain  natural  course  0,.E..O^..O^ 

^i-i  •  •  (?)  cut  it  in  configurations  O ,,  0„.  C.,, .  .  .  (9^.,,  and  if,  besides 
these,  there  are  not  on  it  any  other  kinetic  foci  conju;^ate  to  O, 
between  O  and  Q,  and  no  focus  at  all,  conjugate  to  E,  between 
E  and  Q,  the  action  in  this  natural  course  from  C  to  (9  is  the 
ma.ximum    for  all  courses  O...P^  P^.-.Q;  P   being  a  configura- 

'  111  o  ir  .second  volume  we  hope  to  give  all  necessary  elementary  explanations  on 
this  subject. 

*  Two  courses  are  here  called  not  distinct  if  they  differ  from  one  another  only  in 
the  absolute  mapiiitufle,  not  in  the  proportions,  of  the  conij^onents  of  the  deviations 
by  which  they  difi'er  from  the  standard  course. 


I  r  o  PRELIMINAR  K 

tion  infinitely  nearly  agreeing  with  some  configuration  between  E 
and  <9j  of  the  standard  course  O .  .E ..  6*, , .  O^.. ..  Of_i . .  Q, 
and  0 . . .  F,,  P,."  Q  denoting  the  natural  courses  between  0  and P,, 
and  P^  and  Q,  which  deviate  infinitely  little  from  this  standard  course. 

318.  Considering  now,  for  simplicity,  only  cases  in  which  there 
are  but  two  degrees  (§  165)  of  freedom  to  move,  we  see  that  after 
any  infinitely  small  conservative  disturbance  of  a  system  in  passing 
through  a  certain  configuration,  the  system  will  first  again  pass 
through  a  configuration  of  the  undisturbed  course,  at  the  first  con- 
figuration of  the  latter  at  which  the  action  in  the  undisturbed  motion 
ceases  to  be  a  minimum.  For  instance,  in  the  case  of  a  particle, 
confined  to  a  surface,  and  subject  to  any  conservative  system  of  force, 
an  infinitely  small  conservative  disturbance  of  its  motion  through  any 
point,  O,  produces  a  disturbed  path,  which  cuts  the  undisturbed  path 
at  the  first  point,  O',  at  which  the  action  in  the  undisturbed  path  from 
O  ceases  to  be  a  minimum.  Or,  if  projectiles,  under  the  influence  of 
gravity  alone,  be  thrown  from  one  point,  O,  in  all  directions  with 
equal  velocities,  in  one  vertical  plane,  their  paths,  as  is  easily  proved, 
intersect  one  another  consecutively  in  a  parabola,  of  which  the  focus 
is  O,  and  the  vertex  the  point  reached  by  the  particle  projected 
directly  upwards.  The  actual  course  of  each  particle  from  O  is  the 
course  of  least  possible  action  to  any  point,  P,  reached  before  the 
enveloping  parabola,  but  is  not  a  course  of  minimum  action  to  any 
point,  Q,  in  its  path  after  the  envelop  is  passed. 

319.  Or  again,  if  a  particle  slides  round  along  the  greatest  circle  of 
the  smooth  inner  surface  of  a  hollow  anchor-ring,  the  'action,'  or 
simply  the  length  of  path,  from  point  to  point,  will  be  least  possible 
for  lengths  (§  305)  less  than  nr  Jab.  Thus  if  a  string  be  tied  round 
outside  on  the  greatest  circle  of  a  perfectly  smooth  anchor-ring,  it  will 
slip  off  unless  held  in  position  by  staples,  or  checks  of  some  kind,  at 
distances  of  not  less  than  this  amount,  ttJ ab,  from  one  another  in 
succession  round  the  circle.  With  reference  to  this  example,  see  also 
§  314,  above. 

Or,  if  a  particle  slides  down  an  inclined  hollow  cylinder,  the 
action  from  any  point  will  be  the  least  possible  along  the  straight  path 
to  any  other  point  reached  in  a  time  less  than  that  of  the  vibration 
one  way  of  a  simple  pendulum  of  length  equal  to  the  radius  of  the 
cylinder,  and  influenced  by  a  force  equal  %p  g  cost,  instead  of  ^  the 
whole  force  of  gravity.  But  the  action  will  not  be  a  minimum  from 
any  point,  along  the  straight  path,  to  any  other  point  reached  in  a 
longer  time  than  this.  The  case  in  which  the  groove  is  horizontal 
(;  =  o)  and  the  particle  is  projected  along  it,  is  particularly  simple  and 
instructive,  and  may  be  worked  out  in  detail  with  great  ease,  without 
assuming  any  of  the  general  theorems  regarding  action. 


CHAPTER   III. 

EXPERIENCE. 


820.  By  the  term  Experience,  in  physical  science,  we  designate, 
according  to  a  suggestion  of  Herschel's,  our  means  of  becoming 
acquainted  with  the  material  universe  and  the  laws  which  regulate  it. 
In  general  the  actions  which  we  see  ever  taking  place  around  us  are 
complex,  or  due  to  the  simultaneous  action  of  many  causes.  When, 
as  in  astronomy,  we  endeavour  to  ascertain  these  causes  by  simply 
watching  their  effects,  we  obsen:e;  when,  as  in  our  laboratories,  we 
interfere  arbitrarily  with  the  causes  or  circumstances  of  a  pheno- 
menon, we  are  said  to  ex^er'unent. 

321.  For  instance,  supposing  that  we  are  possessed  of  instru- 
mental means  of  measuring  time  and  angles,  we  may  trace  out  by 
successive  observations  the  relative  position  of  the  sun  and  earth  at 
different  instants;  and  (the  method  is  not  susceptible  of  any  accuracy, 
but  is  alluded  to  here  only  for  the  sake  of  illustration)  from  the 
variations  in  the  apparent  diameter  of  the  former  we  may  "calculate 
the  ratios  of  our  distances  from  it  at  those  instants.  We  have  thus  a 
set  of  observations  involving  time,  angular  position  with  reference  to 
the  sun,  and  ratios  of  distances  from  it;  sufficient  (if  numerous 
enough)  to  enable  us  to  discover  the  laws  which  connect  the  varia- 
tions of  these  co-ordinates. 

Similar  methods  may  be  imagined  as  applicable  to  the  motion  of 
any  planet  about  the  sun,  of  a  satellite  about  its  primary,  or  of  one 
star  about  another  in  a  binary  group. 

322.  In  general  all  the  data  of  Astronomy  are  determined  in  this 
way,  and  the  same  may  be  said  of  such  subjects  as  Tides  and  Meteor- 
ology. Isothermal  Lines,  Lines  of  Equal  Dip  or  Intensity,  Lines  of 
No  Declination,  the  Connexion  of  Solar  Spots  with  Terrestrial  Mag- 
netism, and  a  host  of  other  data  and  phenomena,  to  be  explained 
under  the  proper  heads  in  the  course  of  the  work,  are  thus  deducible 
from  Observation  merely.  In  these  cases  the  apparatus  for  the  gigantic 
experiments  is  found  ready  arranged  in  Nature,  and  all  that  the 
philosopher  has  to  do  is  to  watch  and  measure  their  progress  fo  its 
last  details. 


1 1 2  PRELIMINAR  Y. 

323.  Even  in  the  instance  we  have  chosen  above,  that  of  the 
planetary  motions,  the  observed  effects  are  complex ;  because,  unless 
possibly  in  the  case  of  a  double  star,  we  have  no  instance  of  the 
undisturbed  action  of  one  heavenly  body  on  another;  but  to  a  first 
approximation  the  motion  of  a  planet  about  the  sun  is  found  to  be 
the  same" as  if  no  other  bodies  than-  these  two  existed;  and  the 
approximation  is  sufficient  to  indicate  the  probable  law  of  mutual 
action,  whose  full  confirmation  is  obtained  when,  its  truth  being 
assumed,  the  disturbing  effects  thus  calculated  are  allowed  for,  and 
found  to  account  completely  for  the  observed  deviations  from  the 
consequences  of  the  first  supposition.  This  may  serve  to. give  an 
idea  of  the  mode  of  obtaining  the  laws  of  phenomena,  which  can 
"only  be  observed  in  a  complex  form;  and  the  method  can  always  be 
directly  applied  when  one  cause  is  known  fo  be  pre-eminent. 

324.  Let  us  take  a  case  of  the  other  kind — that  in  which  the  effects 
are  so  complex  that  we  cannot  deduce  fhe  causes  from  the  observation 
of  combinations  arranged  in  ^Nature,  but  must  endeavour  to  form  for 
ourselves  other  combinations  which  may  enable  us  to  study  the  effects 
of  each  cause  separately,  or  at  least  with  only  slight  modification  from 
the  interference  of  other  causes. 

A  stone,  when  dropped,  falls  to  the  ground;  a  brick  and  a  boulder, 
if  dropped  from  the  top  of  a  cliff  at  the  same  moment,  fall  side  by 
side,  and  reach  the  ground  together-  But  a  brick  and  a  slate  do  not; 
and  while  the  former  falls  in  a  nearly  vertical  direction,  the  latter 
describes  a  most  complex  path.  A  sheet  of  paper  or  a  fragment  of 
gold-leaf  presents  even  greater  irregularities  than  the  slate.  But  by 
a  slight  modification  of  the  circumstances,  we  gain  a  considerable 
insight  into  the  nature  of  the  question.  The  paper  and  gold-leaf,  if 
rolled  into  balls,  fall  nearly  in  a  vertical  line.  Here,  fhen,  there  are 
evidently  at  least  two  causes  at  work,  one  which  tends  to  make  all 
bodies  fall,  and  that  vertically;  and  another  which  depends  on  the 
form  and  substance  of  the  body,  and  tends  to  retard  its  fall  and  alter 
its  vertical  direction.  How  can  we  study  the  effects  of  the  former  ^ 
all  bodies  without  sensible  complication  from  the  latter?  The  effects 
of  Wind,  etc.,  at  once  point  out  z^'/ia/ the  latter  cause  i%  the  air  (whose 
^istence  we  may  indeed  suppose  to  have  been  discovered  by  such 
effects);  and  to  study  the  natur'e  of  the  action  of  the  former  it  is 
necessary  to  get  rid  of  the  complications  arising  from  the  presence 
of  air.  Hence  the  necessity  for  Experiment  By  means-  of  an  appa- 
ratus to  be  afterwards  described,  we  remove  the  greater  part  of  the 
air  from  the  interior  of  a  vessel,  and  in  that  we  try  again  our  expe- 
riments on  the  fall  of  bodies;  and  now  a  general  law,  simple  in  the 
extreme,  .though  most  important  in  its  consequences,  is  at  once  appa- 
rent—viz. that  all  bodies,  of  whatever  size,  shape,  .^r  material,  if 
dropped  side  by  side  at  the  same  instant,  fall  side  beside  in  a  space 
void  of  air.  Before  experiment  had  thus  separated  the  phenomena, 
hasty  philosophers  had  rushed  to  the  conclusion  that  some  bodies 
possess  the  quality  of  heaviness^  others  that  of  lightness,  etc.     Had 


EXPERIENCE.  113 

this  state  of  thinsrs  remained,  the  law  of  gravitation,  vigorous  though 

it.s  action  be  throughout  the  universe,  could  never  have  been  recog- 
nized as  a  general  principle  by  the  human  mind. 

Mere  observation  of  lightning  and  its  eflects  couM  never  have  led 
to  the  discovery  of  their  relation  to  the  phenomena  presented  by 
rubbed  amber.  A  modification  of  the  course  of  Nature,  sueh  as  the 
bringing  down  of  atmospheric  electricity  into  our  laboratories,  was 
necessary.  Witliout  exj.criment  we  could  never  even  have  learned 
the  existence  of  terrestrial 

325.  When  a  particular  agent  or  cause  Is  to  be  studied,  experi- 
ments should  be  arranged  in  such  a  way  as  to  lead  if  possible 
to  results  depending  on  it  alone;  or,  if  this  cannot  be  done,  they 
should  be  arranged  so  as  to  shov/  difterences  produced  by  vary- 
ing it. 

325.  Thus  to  determine  the  resistance  of  a  wire  against  the 
conduction  of  electricity  through  it,  we  may  measure  the  whole 
strength  of  current  produced  in  it  by  electromotive  force  betv,-een 
its  ends  when  the  amount  of  this  electrom.otive  force  is  ^iZ'^n, 
or  can  be  ascertained.  But  when  the  wire  is  that  of  a  submarine 
telegraph  cable  there  is  always  an  uiikni'^iun  and  ever  varying 
electromotive  force  between  its  ends,  due  to  the  earth  (produc- 
ing what  is  commonly  called  the  "earth-current"),  and  to  deter- 
mine its  resistance  the  difterence  in  the  strength  of  the  current 
produced  by  suddenly  adding  to  or  subtracting  from  the  terreS' 
trial  electromotive  force,  the  electromotive  force  of  a  given 
voltaic  battery  is  to  be  very  quickly  measured ;  and  this  is  to  be 
done  over  and  over  again,  to  eliminate  the  effect  of  variation  of 
the  earth  current  during  the  few  seconds  of  time  which  must 
elapse  before  the  electro- static  induction  permits  the  current  due  to 
the  battery  to  reach  nearly  enough  its  full  strength  to  p-ractically 
annul  error  on  this  score. 

327.  Endless  patience  and  perseverance  in  designing  and  trying 
different  mctliods  for  investigation  are  necessary  for  the  advancement 
of  science:  and  indeed,  in  discovery,  he  is  the  most  likely  to  succeed 
who,  not  allowing  himself  to  be  di.-jheartened  by  the  non-success  of 
one  form  of  experiment,  judiciously  varies  his  methods,  and  thus 
interrogates  in  every  conceivably  useful  manner  the  subject  of  his 
investigations. 

323.  A  most  important  remark,  due  to  Herschel,  regards  what  are 
called  n'l/t/z/'i?/ phenomena.  ^Vi'-en,  in  an  experiment,  all  known  causes 
being  allowed  for,  there  remain  certain  unexplained  effects  (exces- 
sively sliu'ht  it  may  be),  these  mu:;t  be  carefully  investigated,  and  e\-ery 
conceivable  variation  of  arrana:ement  of  apparatus,  etc.,  tried;  until,  if 
possible,  we  manage  so  to  e.xa^'g.r.i'.e  the  residual  phenomenon  as  to 
be  able  to  detect  its  rau-c.  Ic  \-\  here,  perhaps,  that  in  th.e  p^p^  :r.t 
state  of  science  we  may  mo^t  reasonably  look  for  extensions  of  our 
knowledge  ;  at  all  events  wc  are  warranted  ^y  the  recer.t  !i:;;.o.y  of 
Natural  Philosophy  in  so  doing.     Thus,  to   take   only   a   very  few 


1 1 4  PRELIMINAR  V. 

instances,  and  to  say  nothing  of  the  discovery  of  electricity  and  mag-- 
netism  by  the  ancients,  the  pecuUar  smell  observed  in  a  room  in 
which  an  electrical  machine  is  kept  in  action,  was  long  ago  observed, 
but  called  the  'smell  of  electricity,'  and  thus  left  unexplained.  The 
sagacity  of  Schonbein  led  to  the  discovery  that  this  is  due  to  the 
formation  of  Ozone,  a  most  extraordinary  body,  of  enormous  chem- 
ical energies ;  whose  nature  is  still  uncertain,  though  the  attention  of 
chemists  has  for  years  been  directed  to  it. 

329.  Slight  anomalies  in  the  motion  of  Uranus  led  Adams  and 
Le  Verrier  to  the  discovery  of  a  new  planet ;  and  the  fact  that  a 
magnetized  needle  comes  to  rest  sooner  when  vibrating  above  a 
copper  plate  than  when  the  latter  is  removed,  led  Arago  to  what 
was  once  called  magnetism  of  rotation,  but  has  since  been  explained, 
immensely  extended,  and  applied  to  most  important  purposes.  In 
fact,  this  accidental  remark  about  the  oscillation  of  a  needle  led 
to  facts  from  which,  in  Faraday's  hands,  was  evolved  the  grand 
discovery  of  the  Induction  of  Electrical  Currents  by  magnets  or 
by  other  currents.  We  need  not  enlarge  upon  this  point,  as  in 
the  following  pages  the  proofs  of  the  truth  and  usefulness  of  the 
principle  will  continually  recur.  Our  object  has  been  not  so  much 
to  give  applications  as  methods,  and  to  show,  if  possible,  how  to 
attack  a  new  combination,  with  the  view  of  separating  and  studying 
in  detail  the  various  causes  which  generally  conspire  to  produce 
observed  phenomena,  even  those  which  are  apparently  the  simplest. 

330.  If,  on  repetition  several  times,  an  experiment  continually  gives 
different  results,  it  must  either  have  been  very  carelessly  performed, 
or  there  must  be  some  disturbing  cause  not  taken  account  of  And, 
on  the  other  hand,  in  cases  where  no  very  great  coincidence  is 
likely  on  repeated  trials,  an  unexpected  degree  of  agreement  between 
the  results  of  various  trials  should  be  regarded  with  the  utmost 
suspicion,  as  probably  due  to  some  unnoticed  peculiarity  of  the 
apparatus  employed.  In  either  of  these  cases,  however,  careful 
observation  cannot  fail  to  detect  the  cause  of  the  discrepancies  or 
of  the  unexpected  agreement,  and  may  possibly  lead  to  discoveries 
in  a  totally  unthought-of  quarter.  Instances  of  this  kind  may  be 
given  without  limit;  one  or  two  must  suffice. 

331.  Thus,  with  a  7'ery  good  achromatic  telescope  a  star  appears 
to  have  a  sensible  disc.  But,  as  it  is  observed  that  the  discs  of 
all  stars  appear  to  be  of  equal  angular  diameter,  we  of  course  suspect 
some  common  error.  Limiting  the  aperture  of  the  object-glass 
inorasis  the  appearance  in  question,  which,  on  full  investigation, 
is  found  to  have  nothing  to  do  with  discs  at  all.  It  is,  in  fact,  a  dil- 
fraction  phenomenon,  and  will  be  explained  in  our  chapters  on  Light. 

Again,  in  measuring  the  velocity  of  Sound  by  experiments  con- 
ducted at  night  with  cannon,  the  results  at  one  station  wcrie  never 
found  to  agree  exactly  with  those  at  the  other  ;  sometimes,  indeed, 
the   differences  were  very  considerable.     But  a  little  consideration 


EXPERIENCE.  115 

led  to  the  rf-Tn'-k;  that  on  those  nights  in  which  t^/^  disccr  J;-noe 
was  greatest  a.  strong  wind  was  blowing  nearly  from  one  station 
to  the  other.  Allowing  for  the  obvious  eft'ect  of  this,  or  rather 
eliminating  it  altogether,  the  mean  velocities  on  different  evenings 
v.-ere  found  to  r.gree  very  closely.- 

332.  It  may  perhaps  be  advisable  to  say  a  few  words  here  about 
the  use  of  hypotheses,  and  especially  those  of  very  different  gradations 
of  value  which  are  promulgated  in  the  form  of  Mathematical  Theories 
of  different  branches  of  Natural  Philosophy. 

333.  ^\^lere,  as  in  the  case  of  the  planetary  motions  and  disturb- 
ances, the  forces  concerned  are  thoroughly  known,  the  mathematical 
theory  is  absolutely  true,  and  requires  only  analysis  to  work  out  its 
remotest  details.  It  is  thus,  in  general,  far  ahead  of  observation,  and 
is  competent  to  predict  effects  not  yet  even  observed — as,  for  instance, 
Lunar  Inequalities  due  to  the  action  of  Venus  upon  the  Earth,  etc.  etc., 
to  which  no  amount  of  observation,  unaided  by  theory,  would  ever 
have  enabled  us  to  assign  the  true  cause.  It  may  also,  in  such 
subjects  as  Geometrical  Optics,  be  carried  to  developments  I'ar  beyond 
t'r.e  reach  of  experiment ;  but  in  this  science  the  assumed  bases  of  the 
theory  are  only  approximate,  and  it  fails  to  explain  in  all  their  peculi- 
arities even  such  comparatively  simple  phenomena  as  Halos  and 
Rainbows  ;  though  it  is  perfectly  successful  for  the  practical  purposes 
of  the  maker  of  microscopes  and  telescopes,  and  has,  in  these  cases, 
carried  the  construction  of  instruments  to  a  degree  of  perfection 
which  merely  tentative  processes  never  could  have  reached. 

334.  Another  class  of  mathematical  theories,  based  to  a  certain 
extent  on  experiment,  is  at  present  useful,  and  has  even  in  certain 
cases  I'jointed  to  new  and  important  results,  which  experiment  has 
subsequently  verified.  Such  are  the  Dynamical  Theory  of  Heat,  the 
Undulatory  Theory  of  Light,  etc.  etc.  In  the  former,  which  is  based 
upon  the  cxjieriniLntal  fact  t'lat  heat  is  moiion^  many  Ibrmulae  are 
at  present  obscure  and  uninterpretable,  because  we  do  not  know 
uiJiat  is  mo\ing  or  ho:o  it  rnovLS.  Results  of  the  theory  in  which 
these  are  not  in\-olvcd,  are  of  course  experimentally  verified.  The 
same  difficulties  exist  in  the  Theory  of  Light.  Cut  before  this 
obscurity  can  be  perfectly  cleared  up,  we  must  know  something 
of  the  ultimate,  or  Diolccitlar,  constitution  of  tlie  boJiieS,  or  groups 
of  molecules,  at  present  known  to  us  only  in  the  aggregate. 

335.  A  third  class  is  well  represented  by  the  MathematicalThe'^vks 
of  Heat  (Conduction),  Electricity  (.Statical),  and  Magnetis-n  (Perma- 
nent). Although  we  do  not  know  Jw'i'  Heat  is  ])ropagated  in  l)0'!ies, 
nor  7<:'//j/ Statical  l-Llectricitv  or  Permanent  Magnetism  are,  the  k'ws 
of  their  forces  arc  as  certainly  known  as  that  uf  Gravitation,  and 
can  therefore  like  it  be  developed  to  their  consc-ipjences,  by  ti;e 
application   of    Mathematical   Analysis.     The    works   of    Fourier', 

*  Thbrie  Analy'.ique  de  la  Chalcur.     Paris,  1822. 


1 1 6  PRELIMINAR  V. 

Green",  and  Poisson",  are  renharkable  instances  of  such  develop- 
ment. Another  good  example  is  Ampere's  Theory  of  Electro- 
dynamics. 

336.  Mathematical  theories  of  physical  forces  are,  in  general,  of 
one  of  two  species.  First,  those  in  which  the  fundamental  assump- 
tion is  far  more  general  than  is  necessary.  Thus  the  celebrated 
equation  of  Laplace's  Functions  contains  the  mathematical  foundation 
of  the  theories  of  Gravitation,  Statical  Electricity,- Permanent  Mag- 
netism, Permanent  Flux  of  Heat,  Motion  of  Incompressible  Fluids, 
etc.  etc.,  and  has  therefore  to  be  accompanied  by  limiting  consider- 
ations when  applied  to  any  one  of  these  subjects. 

337.  Again,  there  are  those  which  are  built  upon  a  few  experiments, 
or  simple  but  inexact  hypotheses,  only ;  and  which  require  to  be 
modified  in  the  way  of  extension  rather  than  limitation.  As  a  notable 
example,  we  may  refer  to  the  whole  subject  of  Abstract  Dynamics, 
which  requires  extensive  modifications  (explained  in  Division  III.) 
before  it  can,  in  general,  be  applied  to  practical  purposes. 

338.  When  the  most  probable  result  is  required  from  a  number  of 
observations  of  the  same  quantity  which  do  not  exactly  agree,  we 
must  appeal  to  the  mathematical  theory  of  probabilities  to  guide  us 
to  a  method  of  combining  the  results  of  experience,  so  as  to  eliminate 
from  them,  as  far  as  possible,  the  inaccuracies  of  observation.  But 
it  must  be  explained  that  we  do  not  at  present  class  as  inaccuracies 
cf  observation  any  errors  which  may  affect  alike  every  one  of  a  series 
of  observations,  such  as  the  inexact  determination  of  a  zero-point  or 
of  the  essential  units  of  time  and  space,  the  personal  equation  of  the 
observer,  etc.  The  process,  whatever  it  may  be,  which  is  to  be 
employed  in  the  elimination  of  errors,  is  applicable  even  to  tliese,  but 
only  when  several  distinct  series  of  observations  have  been  made,  with 
a  change  of  instrument,  or  of  observer,  or  of  both. 

339.  We  understand  as  inaccuracies  of  observation  the  whole  class 
of  errors  v.'hich  are  as  likely  to  lie  in  one  direction  as  another  in  suc- 
cessive trials,  and  which  we  may  fairly  presume  would,  on  the  average 
of  an  infinite  number  of  repetitions,  exactly  balance  each  other  in 
excess  and  defect.  Moreover,  we  consider  only  errors  of  such  a 
kind  that  their  probability  is  the  less  the  greater  they  are ;  so  that 
such  errors  as  an  accidental  reading  of  a  wrong  number  of  whole 
degrees  on  a  divided  circle  (which,  by  the  way,  can  in  general  be 
probably  corrected  by  comparison  with  other  observations)  are  not  to 
be  included. 

340.  Mathematically  considered,  the  subject  is  by  no  means  art 
easy  one,  and  many  high  authorities  have  asserted  that  the  reasoning 
employed  by  Laplace,  Gauss,  and  others,  is  not  well  founded  ;  although 
the  results  of  their  analysis  have  been  generally  accepted.  As  an 
excellent  treatise  on  the  subject  has  recently  been  pubhshed  by  Airy, 

'  Essay    on    the  Applicaticni   of   lilath'matical  Analysis  to   the    Thecrrus  of 
Eledricily  and  Magnetism.     Nottingham,  1828.     Reprinted  in  Crelle^s  Journal, 
'  MCmoircs  sur  le  Mugm'tisme.     Man.  de  I'Acad.  des  Sciences,  181 1. 


EXPERTEXCE.  117 

it  is  not  necessary  for  us  to  do  more  than  sketch  in  the  most  cursor}' 
manner  what  is  called  the  Mdhod  of  Least  Squares. 

341.  Supposing  the  zero-point  and  the  graduation  of  an  instrument 
{micrometer,  mural  circle,  thermometer,  clectrom-ctcr,  gah-anomcter, 
etc.)  to  be  absolutely  accurate,  successive  readings  of  the  value  of  a 
quantity  (linear  distance,  altitude  of  a  star,  tciripcrature,  potential, 
strength  of  an  electric  current,  etc.)  may,  and  in  general  do,  con- 
tinually differ.  What  is  most  probably  the  true  value  of  the  observed 
quantity? 

The  most  probable  value,  in  all  such  cases,  if  the  observations  are  all 
equally  reliable,  will  evidently  be  the  simple  mean ;  or  if  they  arc  not 
equ.tlly  reliable,  the  mean  found  by  attributing  iveie-hts  to  the  5e\-e:--\l 
observations  in  proportion  to  their  presumed  exactness.  But  if  several 
such  means  have  been  taken,  or  several  single  observations,  and  if 
these  several  means  or  observations  have  been  differently  qualified 
for  the  determination  of  the  sought  quantity  (some  of  them  being 
likely  to  give  a  more  exact  value  than  others),  we  must  assign  MiV/'c/- 
ically  the  best  method  of  combining  them  in  practice. 

34:2.  Inaccuracies  of  observation  are,  in  general,  as  likely  to  be  in 
excess  as  in  defect.  They  are  also  (as  before  observed)  more  likely 
to  be  srna.ll  than  great ;  and  (practically)  large  errors  are  not  to  be 
expected  at  all,  as  such  would  come  under  the  class  0^ a:\vtial>'c  mis- 
takes. It  follows  that  in  any  one  of  a  series  of  observations  of  the 
same  quantity  the  probability  of  an  error  of  magnitu^le  x,  must  depend 
upon  x^,  and  must  be  expressed  by  some  function  whose  value 
diminishes  very  raj)id!y  as  x  increases.  The  probability  that  the 
error  lies  between  x  and  x-\-hx,  where  hx  is  very  small,  must  also  be 
proportional  to  ex.     The  law  of  error  thus  found  is 

I      _  ■^'  ox 

where  /'  is  a  constant,  indicating  the  degree  of  coarseness  or  delicacy 

of  the  system  of  measurement  employed.     The  co-efficient  — —  secures 

that  the  sum  of  the  probabilities  of  all  possible  errors  shall  be  unity, 
as  it  ought  to  be. 

343.  The  Probable  Error  q(  an  observation  is  a  numerical  quan.titv 
such  that  the  error  of  the  observation  is  as  likely  to  exceed  as  10  fall 
short  of  it  in  m.agnitude. 

If  we  assume  the  law  of  error  just  found,  and  call  P  the  probable 
error  in  one  trial,  v.-e  have  tr.e  approximate  result 

P-~o-j,;-]  h. 

344.  The  probable  error  of  any  given  multiple  of  the  value  of  an 
obscr.'ed  quantity-  is  evidently  the  same  multiple  of  the  probable  error 
of  ihe  quantity  itself 

The  probable  error  of  the  sum  or  ditTerence  of  two  quantiiiL:?, 
aflfected  by  indepcndait  errors,  is  the  square  root  of  the  sum  of  tiie 
squares  of  their  separate  probable  errors. 


1 1 8  PRELIMINAR  Y. 

345.  As  above  remarked,  the  principal  use  of  this  theory  is  in  the 
deduction,  from  a  large  series  of  observations,  of  the  values  of  the 
quantities  sought  in  such  a  form  as  to  be  liable  to  the  smallest  pro- 
bable error.  As  an  instance — by  the  principles  of  physical  astronomy, 
the  place  of  a  planet  is  calculated  from  assumed  values  of  the  elements 
of  its  orbit,  and  tabulated  in  the  Nautical  Almanac.  The  observed 
places  do  not  exactly  agree  with  the  predicted  places,  for  two  reasons 
• — first,  the  data  for  calculation  are  not  exact  (and  in  fact  the  main 
object  of  the  observation  is  to  correct  their  assumed  values);  second, 
the  observation  is  in  error  to  some  unknown  amount.  Now  the 
difference  between  the  observed,  and  the  calculated,  places  depends 
on  the  errors  of  assumed  elements  and  of  observation.  Our  methods 
are  applied  to  eliminate  as  far  as  possible  the  second  of  these,  and  the 
resulting  equations  give  the  required  corrections  of  the  elements. 

Thus  if  h  be  the  calculated  R.A.  of  a  planet :  ha,  he,  8vj,  etc.,  the 
corrections  required  for  the  assumed  elements  :  the  true  R.A.  is 

6  +  Aha  +  £he  +  IISct-  +  etc, 

where  A,  E,  U,  etc.,  are  approximately  known.  Suppose  the  observed 
R.A.  to  be  0,  then 

e  +  ASa  +  £Se  +  Uhz:j  +  ...  =  ©, 
or  A8a  +  Ehe+'n.h'ur+ ...=Q-$, 

a  known  quantity,  subject  to  error  of  observation.  Every  observation 
made  gives  us  an  equation  of  the  same  /orm  as  this,  and  in  general 
the  number  of  observations  greatly  exceeds  that  of  the  quantities  8a, 
he,  8ixr,  etc.,  to  be  found. 

346.  The  theorems  of  §  344  lead  to  the  following  rule  for  com- 
bining any  number  of  such  equations  which  contain  a  smaller  number 
of  unknown  quantities  : — 

Make  the  probable  envr  of  the  second  member  the  same  in  each  equa- 
tion, by  the  e77iployme7it  of  a  proper  factor :  multiply  each  equation  by  the 
co-efficient  of  x  in  it  and  add  all,  for  one  of  the  final  equations  ;  and  so, 
with  reference  to  y,  z,  etc.,  for  the  others.  The  probable  errors  of  the 
■\?alues  of  X,  y,  etc.,  found  from  these  final  equations  will  be  less  than 
those  of  the  values  derived  from  any  other  linear  method  of  com- 
bining the  equations. 

This  process  has  been  called  the  method  of  Least  Squares,  because 
the  values  of  the  unknown  quantities  found  by  it  are  such  as  to  render 
the  sum  of  the  squares  of  the  errors  of  the  original  equations  a 
minimum. 

347.  "\\Tien  a  series  of  observations  of  the  same  quantity  has  been 
made  at  different  times,  or  under  different  circumstances,  the  law 
conng:ting  the  value  of  the  quantity  with  the  time,  or  some  other 
variable,  may  be  derived  from  the  results  in  several  ways — all  more 
or  less  approximate.  Two  of  these  methods,  however,  are  so  much 
more  extensively  used  than  the  others,  that  we  shall  devote  a  page  or 


EXPERFEXCE.  119 

two  here  to  a  preliminary  notice  of  them,  leaving  detailed  instances 
of  their  application  till  we  come  to  Heat,  Klcctncity,  etc.  Th(iy 
consist  in  (i)  a  Curve,  giving  a  graphic  representation  of  the  relation 
between  the  ordinate  and  abscissa,  and  (2)  an  Empirical  Fonnula 
cunnecting  the  variables. 

348.  Thus  if  the  abscissae  represent  intervals  of  time,  and  the 
crciinates  tlie  corresponding  height  of  the  barometer,  we  mav  con- 
struct curves  which  show  at  a  glance  the  dependence  of  Ijaroiiictric- 
pressure  up'On  the  time  of  day  ;  and  so  on.  Such  cur\-es  may  l^e 
accurately  drawn  by  jihotographic  processes  on  a  sheet  of  sensitive 
paper  placed  behmd  the  mercurial  column,  and  made  to  m.ove  past 
it  with  a  uniform  horizontal  velocity  by  clockwork.  A  similar  pio- 
cess  is  ap[jlied  to  the  Teiiipera'ure  and  Electricity  of  the  atmosphere, 
and  to  the  components  of  terrestrial  magnetism. 

349.  \\'hcn  the  observations  are  not,  as  in  tiie  last  section,  con- 
tinuous, they  give  us  only  a  series  of  points  in  the  curve,  from  wiii<-h, 
however,  we  may  in  general  approximate  very  closely  to  the  result 
of  continuous  observation  by  drawing.  Iib'-rj  nia'iu,  x  cur\-e  passing 
through  these  points.  This  process,  howe\-er,  must  be  employed 
with  great  caution  ;  because,  unless  the  observations  are  sufheiently 
close  to  each  othccr,  most  important  iluctuations  m  the  curve  may 
escape  notice.  It  is  applicable,  with  abundant  accuracy,  to  all  cases 
where  the  r[uantity  observed  changes  very  slowly.  Thus,  for  instance, 
weekly  observations  of  the  temperature  at  depths  of  from  6  to  2\  feet 
underground  were  found  by  Forbes  suflicient  for  a  very  accurate 
approximation  to  the  law  of  the  phenomenon. 

350.  As  an  instance  of  tlie  processes  employed  for  obtaining  an 
empirical  formula,  we  may  mention  metliods  of /'//'.■r/'('/,?//(V/.  to  wlneh 
the  problem  can  always  be  reduced.  Thus  from  sextant  olnervations, 
at  known  intervals,  of  the  altitude  of  the  sun,  it  is  a  commvm  problem 
of  Astronomy  to  determine  at  what  iiistant  the  altitude  is  greatest, 
and  what  is  that  greatest  altitude.  The  tlrst  enables  us  to  rind  tlie 
true  sol-ir  time  at  the  place,  and  th.e  second,  by  the  lielp  of  the 
Nautical  Almanac,  gives  the  latitude.  Tlie  calculus  of  unite  ditier- 
entes  gives  us  formulae  proper  for  various  data  ;  and  Lagrange  has 
shown  how  to  obtain  a  very  useful  one  by  elementary  algebra. 

In  finite  differences  we  have 

f{x  +  h)  =/(.v)  +  /.  V(-V)  -r  ^-^^  y/i-X)  4-  .  .  . 

This  is  useful,  inasmuch  as  the  successive  differences,  A/!'.v\ 
Ay  (a-),  etc.,  are  easily  calculated  from  the  tabulated  results  of  obser- 
vation, provided  these  have  been  taken  for  cviual  successive  m- 
crements  of  .v. 

If  for  values  .r^,  .v, v„,  a  function  takes  the  values  \\,  j'_,,  J'y- 

}\,  Lagrange  gi\-es  for  it  the  obvious  expression 


Vj 


£   (x-j -/;j){xj- jTj  )...;,.(■, -x„)       x-x„(x.^-x^)\.c.i-x.^...\r.y 


.]ij:-xp  \x- 


T20  ,         PRELIMINARY. 

Here  Fs  assumed  that  the  function  required  is  a  rational  and 
integral  one  in  x  of  the  «— i'**  degree;  and,  in  general,  a  similar 
limitation  is  in  practice  applied  to  the  other  formula  above ;  for  in 
order  to  find  the  complete  expression  for  fix),  it  is  necessary  to 
determine  the  values  of  A/(.t),  L^-f{x),....  If  n  of  the  co-efficients  be 
required,  so  as  to  give  the  n  chief  terms  of  the  general  value  oifix), 
we  must  have  n  observed  simultaneous  values  of  x  and/(x),  and  the 
expression  becomes  determinate  and  of  the  n  -  i**'  degree  in  h. 

In  practice  it  is  usually  sufficient  to  employ  at  most  three  terms 
of  the  first  series.  Thus  to  express  the  length  /  of  a  rod  of  metal  as 
depending  on  its  temperature  t,  we  may  assume 

/g  being  the  measured  length  at  any  temperature  t^i  A  and  B  are  to 
be  found  by  the  method  of  least  squares  from  values  of  /  observed  for 
different  given  values  of  /. 

351.  These  formulae  are  practically  useful  for  calculating  the 
probable  values  of  any  observed  element,  for  values  of  the  in- 
dependent variable  lying  within  the  range  for  v/hich  observation  has 
given  values  of  the  element.  But  except  for  values  of  the  inde- 
pendent variable  either  actually  within  this  range,  or  not  far  beyond 
it  in  either  direction,  these  formulae  express  functions  which,  in 
general,  will  differ  more  and  more  widely  from  the  truth  the  further 
their  application  is  pushed  beyond  the  range  of  observation. 

In  a  large  class  of  investigations  the  observed  element  is  in  its 
nature  a  periodic  function  of  the  independent  variable.  The  har- 
monic analysis  (§  88)  is  suitable  for  all  such.  When  the  values  of  the 
independent  variable  for  vvhich  the  element  has  been  observed  are 
not  equidifferent  the  co-efficients,  determined  according  to  the  method 
of  least  squares,  are  found  by  a  process  which  is  necessarily  very 
laborious ;  but  when  they  are  equidifferent,  and  especially  when  the 
difference  is  a  submultiple  of  the  period,  the  equation  derived  from 
the  method  of  least  squares  becomes  greatly  simplified.  Thus,  if  Q 
denote  an  angle  increasing  in  proportion  to  /,  tlie  time,  through  four 
right  angles  in  the  period,  T,  of  the  phenomenon;  so  that 

2rct 

let  /(0)-^^-K^jCos6>4-^jCOS26i+... 
-I- i?,  sin  ^ -f- ^g  sin  2(? -^  ... 
where  A^,  A^,  A^,...B^^  -^j,---  are  unknown  co-efficients,  to  be 
determined  so  that /(5)  may  express  the  most  probable  value  of  the 
element,  not  merely  at  times  between  observations,  but  through  all 
time  as  long  as  the  phenomenon  is  strictly  periodic.  By  taking  as 
many  of  these  coefficients  as  there  are  of  distinct  data  by  observation, 
the  formula  is  made  to  c.gree  precisely  with  these  data.  But  in  most 
applications  of  the  method,  the  periodically  recurring  part  of  the  phe- 
nomenon is  expressible  by  a  small  number  of  terms  of  the  harmonic 
series,  and  the  higher  terms,  calculated  from  a  great  number  of  data, 


EXPERIENCE.  i2\ 

express  either  irregularities  of  the  phenomenon  not  likely  to  recur: 
or  errors  of  observation.  Thus  a  comparatively  small  number  of 
terms  may  give  values  of  the  element  even  fur  the  very  times  of  ob- 
servation, more  probable  than  the  values  actually  recorded  as  having 
been  observed,  if  the  observations  are  numerous  but  not  mmutely 
accurate. 

The  student  may  exercise  himself  in  writing  out  the  equations  to 
determine  five,  or  seven,  or  more  of  the  coefficients  according  to  the 
method  of  least  squares ;  and  reducing  them  by  proper  formulae  of 
analytical  trigonometry  to  their  simplest  and  most  easily  calculated 
forms  where  the  values  of  B  for  which /(6)  id  given  are  equidifferent. 

27r 

He  will  thus  see  that  when  the  difference  is  -.'-,  i  being  any  integer, 

and  when  the  number  of  the  data  is  i  or  any  nvaltiple  of  it,  the  equa- 
tions contain  each  of  them  only  one  of  the  unknown  quantities  :  so 
that  the  method  of  least  squares  affords  the  most  probable  values  of 
the  co-efficients,  by  the  easiest  and  most  direct  elimination. 


CHAPTER   IV. 

MEASURES  AND   INSTRUMENTS. 


352.  Having  seen  in  the  preceding  chapter  that  for  the  investiga- 
tion  of  the  laws  of  nature  we  must  carefully  watch  experiments,  either 
those  gigantic  ones  which  the  universe  furnishes,  or  others  devised 
and  executed  by  man  for  special  objects — and  having  seen  that  in 
all  such  observations  accurate  measurements  of  Time,  Space,  Force, 
etc,  are  absolutely  necessary,  we  may  now  appropriately  describe  a 
few  of  the  more  useful  of  the  instruments  employed  for  these  pur- 
poses, and  the  various  standards  or  units  which  are  employed  in 
them. 

353.  Before  going  into  detail  we  may  give  a  rapid  resumk  of  the 
principal  Standards  and  Instruments  to  be  described  in  this  chapter. 
As  most,  if  not  all,  of  them  depend  on  physical  principles  to  be 
detailed  in  the  course  of  this  work,  we  shall  assume  in  anticipation 
the  establishment  of  such  principles,  giving  references  to  the  future 
division  or  chapter  in  which  the  experimental  demonstrations  are 
more  particularly  explained.  This  course  will  entail  a  slight,  but 
unavoidable,  confusion — slight,  because  Clocks,  Balances,  Screws, 
etc.,  are  familiar  even  to  those  who  know  nothing  of  Natural  Phi- 
losophy ;  unavoidable,  because  it  is  in  the  very  nature  of  our  subject 
that  no  one  part  can  grow  alone,  each  requiring  for  its  full  develop- 
ment the  utmost  resources  of  all  the  others.  But  if  one  of  our 
departments  thus  borrows  from  others,  it  is  satisfactory  to  find  that 
it  more  than  repays  by  the  power  which  its  improvement  affords 
them. 

354.  We  may  divide  our  more  important  and  fundamental  instru- 
ments into  four  classes — 

Those  for  measuring  Time ', 

„  „  Space,  linear  or  angulaj- ; 

„  „  Force ; 

„  ,,  Mass. 

Other  instruments,  adapted  for  special  purposes  such  as  the 
measurement  of  Temperature,  Light,  Electric  Currents,  etc.,  will 
come  more  naturally  under  the  head  of  the  particular  physical 
energies  to  whose  measurement  they  are  applicable.     Descriptions  of 


MEASURES  AND  INSTRUMENTS.  123 

self-recording  instruments  such  as  tide-gauges,  and  barometers,  ther- 
mometers, electrometers,  recording  photographically  or  otherwise  the 
continuously  varying  pressure,  temperature,  moisture,  electric  poten- 
tial of  the  atmosphere,  and  magnetometers  recording  photographi- 
cally the  continuously  varying  direction  and  magnitude  of  the  terres- 
tricTl  magnetic  force,  must  likewise  be  kept  for  their  proper  places  in 
our  work. 

Calculating  Machines  have  also  important  uses  in  assisting 
physical  research  in  a  great  variety  of  ways.  They  belong  to  two 
classes: — 

I.  Purely  Arithmetical,  dealing  with  integral  numbers  of  units. 
All  of  this  class  arc  evolved  from  the  primitive  use  of  the  calculuses 
or  little  stones  for  counters  (from  which  are  derived  the  very  names 
calndation  and  "  The  Calculus  "),  through  such  mechanism  as  that 
of  the  Chinese  Abacus,  still  serving  its  original  purpose  well  in 
infant  schools,  up  to  the  Arithmometer  of  Thomas  of  Colmar  and  the 
grand  but  partially  realized  conceptions  of  calculating  machines  by 
Babbage. 

II.  Continuous  Calculating  Machines.  These  are  not  only  useful 
as  auxiliaries  for  physical  research  but  also  involve  important  dy- 
nam.ical  and  kinematical  principles  belonging  properly  to  our  subject. 

355.  We  shall  now  consider  in  order  the  more  prominent  instru- 
ments of  each  of  these  four  classes,  and  some  of  their  most  important 
applications  : — 

Clock,  Chronometer,  Chronoscope,  Applicatiofts  to  Observation 

and  to  self-registering  Instruments. 
Vernier   and   Screw-Micrometer,    Cathetometer,    Spherometer, 

Dividing  Engine,  Theodolite,  Sextant  or  Circle. 
Common  Balance,  Bifilar  Balance,  Torsion  Balance,  Pendulum, 

Dynamometer. 

Among  Standards  we  may  mention^ 

1.  Tiine. — Day,  Hour,  Minute,  Second,  sidereal  and  solar. 

2.  Space. — Yard  and  Metre:  Radian,  Degree,  Minute,  Second. 

3.  Farce. — Weight  of  a  Pound  or  Kilogramme,  etc.,  in  any  par- 

ticular locality  (gravitation  unit);  poundal  or  dyne,  Kinetic 
Unit 

4.  Mass. — Pound,  Kilogramme,  etc. 

356.  Although  without  instruments  it  is  impossible  to  procure  or 
apply  any  standard,  yet,  as  without  the  standards  no  instrument  could 
give  us  absolute  measure,  we  may  consider  the  stanciartis  first  — 
referring  to  the  instruments  as  if  we  already  knew  their  principles 
and  applications. 

357.  First  we  may  notice  the  standards  or  units  of  angular 
tpeasure : 


1 24  PRELIMINAR  V. 

Radiati^  or  angle  whose  arc  is  equal  to  radius; 

Degree,  or  ninetieth  part  of  a  right  angle,  and  its  successive 
subdivisions  into  sixtieths  called  Minutes,  Seconds,  Thirds,  etc.  The 
division  of  the  right  angle  into  90  degrees  is  convenient  because  it 
makes  the  half-angle  of  an  equilateral  triangle  (sm"'  i)  an  integral 
number  (30)  of  degrees.  It  has  long  been  universally  adopted  by  all 
Europe.  The  decimal  division  of  the  "right  angle,  decreed  by  the 
fVench  Republic  when  it  successfully  mtroduced  other  more  sweeping 
changes,  utterly  and  deservedly  failed. 

The  division  of  the  degree  into  60  minutes  and  of  the  minute  into 
60  seconds  is  not  convenient;  and  tables  of  the  circular  functions  for 
degrees  and  hundredths  of  the  degree  are  much  to  be  desired. 
Meantime,  when  reckoning  to  tenths  of  a  degree  suffices  for  the 
accuracy  desired,  in  any  case  the  ordinary  tables  suffice,  as  6'  is  ^V  of 
a  degree. 

The  decimal  system  is  exclusively  followed  in  reckoning  by  radians. 
The  value  of  two  right  angles  in  this  reckoning  is  3-14159...  ,  or  it. 
Thus  7r  radians  is  equal  to  180°.  Hence  i8o"^7ris  57"-29578  ... , 
or  57°  17'  44"-8  is  equal  to  one  radian.  In  mathematical  analysis, 
angles  are  uniformly  reckoned  in  terms  of  the  radian. 

358.  The  practical  standard  of  time  is  the  Siderial  Day,  being  the 
period,  nearly  constant',  of  the  earth's  rotation  about  its  axis  (§  237). 
From  it  is  easily  derived  the  Mean  Solar  Day,  or  the  mean  interval 
which  elapses  between  successive  passages  of  the  sun  across  the 
meridian  of  any  place.  This  is  not  so  nearly  as  the  former,  an  abso- 
lute or  invariable  unit;  secular  changes  in  the  period  of  the  earth's 

*  In  our  first  edition  of  our  larger  treatise  it  was  stated  that  Laplace  had  calculated 
from  ancient  observations  of  echpses  that  tlie  period  of  the  earth's  rotation  about 
its  axis  had  not  altered  by  rrmiir^-^  of  itself  since  720  B.C.  In  §  830  it  was 
pointed  out  that  this  conclubiou  is  overthrown  by  farther  information  from 
I'hysical  Astronomy  acquired  in  the  interval  between  tlie  printmg  of  the  two 
sections,  in  virtue  of  a  correction  whicii  Adams  had  made  as  early  as  1863  upon 
Laplace's  dynamical  investigatiori  of  an  acceleration  of  the  moon's  mean  motion, 
j)roduced  by  the  Sun's  attraction,  showing  that  only  about  half  of  the  observed 
acceleration  of  the  moon's  mean  motion  relatively  to  the  angular  velocity  of  the 
earth's  rotation  was  accounted  for  by  this  cause.  [Quoting  from  the  first  edition, 
S830.]  "In  1859  Adams  communicated  to  Delaunay  his  final  result: — that  at 
•"the  end  of  a  century  -the  moon  is  5" "7  before  the  position  she  would  have, 
"relatively  to  a  meridian  of  the  earth,  according  to  the  angular  velocities  of  the 
"two  motions,  at  the  beginning  of  the  century,  and  the  acceleration  of  the 
"moon's  motion  truly  calculated  from  the  various  disturbing  causes  then  recog- 
"nized.  Delaunay  soon  after  verified  this  result;  and  about  the  beginning  of 
"1866  suggested  that  the  true  explanation  may  be  a  retardation  of  the  earth's 
•oiation  by  tidal  friction.  Using  tins  hypothesis,  and  allowing  for  the  conse- 
■'quent  retardation  of  the  moon's  mean  motion  by  tidal  reaction   (§  276),  Adams, 

'in  an  estimate  which  he  has  commurucated  to  us,  founded  on  the  rough  as- 
"sumpiion  that  the  parts  of  the  earth's  retardation  due  to  solar  and  lunar  tides 

"are  as  the  sc[uares  of  the  respective  tide-generating  forces,  finds  ■22'  as  the 
•'error  by  which  the  earth  would  in  a  century  get  behind  a  perfect  clock  rated 
"at  the  beginning  of  the  centui7.  If  the  retardation  of  rate  giving  this  integral 
"effect  were  uniform  (§  32),  the  earth,  as  a  timekeeper,  would  be  going  slower 
'"by  ••22  of  a  secontl  per  year  in  the-  middle,  or  ^44  of  a  second  per  yearat  the 
"end,  than  at  the  beginning  of  a  century." 


MEASURES  AND   IXSTRIWEXTS.  125 

revolution  round  the  sun  affect  it,  though  very  sh'ghtly.  It  is  divitlecl 
into  24  hours,  and  the  hour,  hke  the  tlcgrce,  is  subdivided  uito 
successive  sixtieths,  called  minutes  ami  seconds.  I'r.c  usual  sub- 
division of  seconds  is  decimal. 

It  is  well  to  observe  that  seconds  and  minutes  o,{  time  are  distin- 
guished from  those  of  angular  measure  by  notation.  Thus  we  have 
for  time  13'' 43"  2 7 '-5 8,  but  for  angular  measure  13"  43' .:?' '5  x 

When  long  periods  of  tune  are  to  be  measured,  tiie  !::ean  ^o'ar 
year,  consisting  of  366-242203  sidenal  days,  or  365-242242  mean 
solar  days,  or  the  century  consisting  of  100  such  years,  may  be  con- 
veniently employed  as  the  unit. 

359.  The  ultimiate  standard  of  accurate  chronometry  must  (if  the 
human  race  live  on  the  earth  for  a  few  million  years)  be  founded  on 
the  physical  properties  of  some  body  of  more  constant  character 
than  the  earth  for  instance,  a  carefully-arranged  metallic  spring, 
hermetically  sealed  in  an  exhausted  glass  vessel.  The  time  of  vibra- 
tion of  such  a  spring  would  be  necessarily  more  constant  from  day  to 
day  than  that  of  the  balance-s])nng  of  the  best  possible  chronometer, 
disturbed  as  this  is  by  the  train  of  mechanism  with  which  it  is  con- 
nected: and  It  would  certainly  be  more  constant  from  age  to  age 
than  the  time  of  rotation  of  the  earth,  retarded  as  it  now  is  by  tidal 
resistance  to  an  extent  that  becomes  very  sensible  in  2000  years  , 
and  cooling  and  shrinking  to  an  extent  tliat  must  produce  a  \ery 
considerable  effect  on  its  time-keeping  in  fil'ty  million  years. 

360.  The  British  standard  of  length  is  the  Imperial  Yard,  derlned 
as  the  distance  between  two  marks  on  a  certain  metallic  bar,  pre- 
served in  the  Tower  of  London,  when  the  whole  lias  a  temperature  of 
60''  Fahrenheit.  It  was  not  directly  derived  from  any  fixed  cjuantity 
in  nature,  although  some  important  relations  witli  natural  elements  ha\e 
been  measured  with  gTeat  accuracy.  It  lias  been  carefully  comi>ared 
with  the  length  of  a  second's  pendulum  vi!)rating  at  a  certain  station  in 
the  neighbourhnod  of  London,  so  that  shoulil  it  again  be  destroyed. 
as  It  was  at  the  burning  of  the  Houses  of  Parliament  m  1S31,  .uid 
should  all  exact  copies  (jf  it,  of  which  several  are  pre^^erved  in  winuus 
places,  be  also  lost,  it  can  be  restored  by  pendulum  observatu'iis.  .\ 
less  accurate,  but  still  (unle-s  m  the  event  of  earthquake  di-^turbance) 
a  very  good,  iiie;ins  of  reproduring  it  exists  m  tiie  measured  base-lines 
of  the  Ordnance  .Survey,  and  the  thence  calculated  distances  between 
definite  station^  111  tlie  Uritisii  Islands,  winch  lia\e  been  ascertaiiieil 
m  terms  of  it  with  a  d^i^ree  of  acciirary  sometimes  witJiin  an  mc'n 
per  mile,  tliat  is  to  say,  uitlun  about  ^-pijuu- 

361.  In  scientiiic  investigations,  we  endeavour  as  much  as  ])Ossib!e 
10  keep  to  one  unit  at  a  time,  and  the  f 'Ot,  whieli  is  defined  t.- ■  be 
one-third  part  of  the  yard,  is,  lur  iJntish  measurement,  generally 
adopted.  Unfurtunately  the  inch,  or  oiie-tueltth  of  a  foot,  ma>!; 
sometimes  be  used,  but  it  is  subdi\'ided  de<  imally.  The  statr.te  in;'e, 
or  1760  yards,  is  unfortunately  olten  used  when  great  lengths  on  bmd 


126  PRELIMINARY, 

are  considered;  but  the  sea-mile,  or  average  minute" of  latitude,  is 
much  to  be  preferred.  Thus  it  appears  that  the  British  measurement 
of  length  is  more  inconvenient  in  its  several  denominations  than  the 
European  measurement  of  time,  or  angles. 

362.  In  the  French  metrical  system  the  decimal  division  is  exclu- 
sively employed.  The  standard,  (unhappily)  called  the  Metre,  was 
defined  originally  as  the  ten-millionth  part  of  the  length  of  the 
quadrant  of  the  earth's  meridian  from  the  pole  to  the  equator;  but  it 
is  now  defined  practically  by  the  accurate  standard  metres  laid  up  in 
various  national  repositories  in  Europe.  It  is  somewhat  longer  than 
the  yard,  as  the  following  Table  shows  : 


Centimetre  =    "3937043  inch. 

Ml'tre  =  3-280869  feet. 
Kilometre  =    •6213767  British 
Statute  mile. 


Inch  ^  25*39977  millimetres. 
Foot=  3-047972  decimetres. 
British  Statute  mile 

=  1609-329  metres. 

363.  The  unit  of  superficial  measure  is  in  Britain  the  square  yard, 
in  France  the  mfetre  carrd.  Of  course  we  may  use  square  inches, 
feet,  or  miles,  as  also  square  millimetres,  kilometres,  etc.,  or  the 
Hectare  =  10,000  square  metres. 

Square  inch  =     .6-451483  square  centimetres. 
,,      foot=      9-290135       „      decimetres. 
„      yard=    83-61121         „      decimetres. 
Acre  =       -4046792  of  a  hectare. 

Square  British  Statute  mile  =  258-9946  hectare. 
Hectare  =      2-471093  acres. 

364.  Similar  remarks  apply  to  the  cubic  measure  in  the  two 
countries,  and  we  have  the  following  Table : — 

Cubic  inch  =    16-38661  cubic  centimetres. 

„      foot=    28-31606     „     decimetres  or  Zw'r^J. 
Gallon        =     4-543808  litres. 

„  =277-274  cubic  inches,  by  Act  of  Parliament, 

now  repealed. 
Litre  =    '035315  cubic  feet. 

365.  The  British  unit  of  mass  is  the  Pound  (defined  by  standards 
only);  the  French  is  the  Kilogramme,  defined  originally  as  a  litre  of 
water  at  its  temperature  of  maximum  density;  but  now  practically 
defiTied  by  existing  standards. 

Grain  =64-79896.  milligrammes.  I  Gramme     =  15-43235  grains. 
Pound  =  453-5927  grammes.  [  Kilogram.  =  2-20362125  lbs. 

Professor  W.  H.  Miller  finds  {Phil.  Trans.,  1857)  that  the  '  kilo- 
gramme  des  Archives'  is  equal  in  mass  to  15432-349  grains:  and 
the  '■kilogramme  type  laiton^  deposited  in  the  Ministere  de  ITnttfrieure 
in  Paris,  as  standard  for  French  commerce,  is  15432*344  grains. 

366.  The  measurement  of  force,  whether  in  terms  of  the  weight 
of  a  stated  mass  in  a  stated  locality,  or  in  terms  of  the  absolute  or 


MEASURES  AND   IXSTRUMEXTS.  127 

kinetic  unit,  has  been  explained  in  Chapter  II.  (Sec  §,?22i — 227.) 
YxQ\w  the  measures  of*  force  and  length  we  derive  at  once  the  measure 
of  work  or  iv.echanical  effect.  That  practically  emidoved  by  engi- 
neers is  founded  on  the  gravitation  measure  of  force.  Neglecting  the 
difference  of  gravity  at  London  and  Paris,  we  see  from  the  above 
Tables  that  the  following  relations  e.xist  between  the  London  and  the 
Pariiian  reckoning  of  work:— 

Foot-pound  =0-13825  kilogramme-metre. 

Kilogramme-metre  =7'233i  foot-pounds. 

357.  A  Clock  is  primarily  an  instrument  which,  by  means  of  a 
train  of  wheels,  records  the  number  of  vibrations  executed  by  a 
pendulum  ;  a  Chro7uvncter  or  Watch  performs  the  same  duty  for  the 
oscillations  o[  a  fiat  spiral  spring — just  as  the  train  of  wheel-work  in 
a  gas-meter  counts  the  number  of  revolutions  of  the  main  shaft 
caused  by  the  passage  of  the  gas  through  the  machine.  As,  how- 
ever, it  is  impossible  to  avoid  friction,  resistance  of  air,  etc.,  a  penda= 
lum  or  spring,  left  to  itself,  would  not  long  continue  its  oscillations, 
and,  while  its  miction  continued,  would  perform  each  oscillation  in 
less  and  less  time  as  tlio  arc  of  ^■ibration  diminished  :  a  continuous 
supply  of  energy  is  furnished  by  the  descent  of  a  v.-eight,  or  the 
uncoihng  of  a  powerful  spring.  This  is  so  applied,  through  the 
train  of  wheels,  to  the  pendulum  or  balance-wheel  by  means  of  a 
mechanical  contrivance  called  an  Escapc/ner.t,  that  the  oscillations  are 
maintained  of  nearly  uniform  extent,  and  therefore  of  nearly  uniform 
duration.  The  construction  of  escapernents,  as  well  as  of  fains  of 
clock-wheels,  is  a  matter  of  Mechanics,  with  the  details  of  which  we 
are  not  concerned,  although -it  may  easily' be  made,  the  subject  ol 
mathematical  investigation.  The  means  of  avoiding  errors  intro- 
duced by  changes  of  temperature,  which  have  been  carried  out  in 
Coinpcnsatioti  pendulums  and  balances,  will  be  more  propcriy  descriljcd 
in  our  chapters  on  Heat.  It  is  to  be  obser\-cd  tint  there  is  lirtle 
inconvenience  if  a  clock  lose  or  gain  re^^ularly ;  that  can  be  easily 
and  accurately  allowed  for  :  irregular  rate  is  fatal. 

368.  By  means  of  a  recent  application  of  electriciiy,  to  be  after- 
wards described,  one  good  clock,  carefully  regulated  from  time  to 
time  to  agree  with  astronomical  observations,  may  be  made  (wiiliout 
injury  to  its  own  performance)  to  control  any  number  of  other  le-s- 
perfectly  constructed  clocks,  so  as  to  compel  their  pendulums  to 
vibrate,  beat  for  beat,  with  its  own. 

369.  In  astronomical  observations,  time  is  estimated  to  tenths  of 
a  second  by  a  practised  ob^-erver,  who,  while  v,-atching  the  j-he- 
nomena,  counts  the  beats  of  the  clock.  Lut  for  the  7'cry  accur.ite 
measun;ment  of  short  iiUervals,  many  in-truments  have  been  devi'-ed. 
Thus  if  a  small  orifice  be  opened  in  a  large  and  deep  ves^^el  fi'i!  '->f 
mercury,  and  if  we  know  bv  trial  the  weight  of  metal  that  es'ipes 
say  in  five  minutes,  a  simple  proportion  gives  the  interval  which 
elapses  during  the  escape  of  any  given  weight      It  is  easy  to  con- 


128 


FMELIMINARY. 


trive  an  adjustment  by  whidh  a  vessel  may  be  pl^ed  under,  and 
withdrawif  frofli,  the  issuing  sVeam  at  the. time  of  occurrence  of  any 
two  successive  phenomena. 

370.  Other  contrivances  are  sometimes  employed,  called  Stop- 
watches, Chronoscopes,  etc.,  whic;h_5;an  be  read  off  at  rest,  started 
on  the  occurrence  of  any  phenomenon,  and  stopped  at  the  oc- 
currence of  a  second,  then  again  read  off;  or  which  allow  of  the 
making  (by  pressing  a  stud)  a  slight  ink-mark,  on  a  dial  revolving 
at  a  given  rate,  at  the  instant  of  the  occurrence  of  each  phe- 
nomenon to  be  noted.  But,  of  late,  these  have  almost  entirely  given 
place  to.  the  Electric  Chronoscope,  an  instrument  which  will  be  fully 
described  later,  when  we  shall  have  occasion  to  refer  to  experiments 
in  which  it  has  been  usefully  employed. 

37X.  We  now  come  to  the  measurement  of  space,  and  of  angles, 
and  for  these  purposes  the  most  important  instruments  are  the  Vernier 
and  the  Screw. 

372.  Elementary  geometry,  indeed,  gives  us  the  means  of  dividing 
any  straight  line  into  any  assignable  number  of  equal  parts;  but  in 

practice  this  is  by  no  means  an  accurate 
or  reliable  method.  It  was  formerly  used 
in  the  so-called  Diagonal  Scale,  of  which 
the  construction  is  evident  from  the  dia- 
gram. The  reading  is  effected  by  a 
sliding  piece  whose  edge  is  perpendicular 
to  the  length  of  the  scale.  Suppose  that 
it  is  PQ  whose  position  on  the  scale  is 
required.  This  can  evidently  cut  only  one 
of  the  transverse  lines.  Its  number  gives 
the  number  of  tenths  of  an  inch  (4  in  the 
figure),  and  the  horizontal  line  next  above 
the  point  of  intersection  gives  evidently 
the  number  of  hundredths  (in  the  present  ca^e  4).  Hence  the 
reading  is  7  "44.  As  an  idea  of  the  comparative  uselessness  of 
this  method,  we  may  mention  that  a  quadrant  of  3  feet  radius, 
which  belonged  to  Napier  of  Merchiston,  and  is  divided  on  the 
limb  by  this  method,  reads  to  minutes  of  a  degree;  no  higher 
accuracy  than  is  now  attainable  by  the  pocket  sextants  made  by 
Troughton  and  Simms,  the  radius  of  whose  arc  is  virtually  litde 
more  than  an  inch.  The  latter  instrument  is  r§ad  by  the  help 
of  a  Vernier. 

373.  The  Vernier  is  commonly  employed  for  such  Instruments  as 
the  Barometer,  Sextant,  and  Cathetometer,  while  the  Screw  is  applied 
to  the  more  delicate  instruments,  such  as  Astronomical  Circles, 
Micrometers,  and  the  Spherometer. 

374.  The  vernier  consists  of  a  slip  of  metal  which  slides 
along  a  divided  scale,,  the  edges  of  the  two  being  coincident. 
Hence,  when  it  is  applied  to  a  divided  circle,  its  edge  is  circular, 


MTASURJtS  AND   /XSTR  iWfEXTS. 


I2> 


■  n  t:ic  vern;er, 
h'.v.\)   or  scjIc 


•ii) 


\- 


niid    it    moves    al^out    an    axis    passing  throuirh  the    centre    of   the 

diviJed    liinl). 

In  the  sketch  let  0,  1,  2,  ...  10  denote  th.e  di\iMom  • 
c.  I.  :,  etc,  any  set  of  consecutive  divisions  on  itic 
a!o;i^  wlii.-c  t.d^e  it  slides.  li,  when  0  and  o  coin- 
cide, lO  and  II  coincide  also,  then  lo  diviMons  of 
the  vcrriicr  are  e(iua!  in  ii.n-;;h  to  ii  on  t.he  hndj; 
and  tiiercfore  eai  h  division  of  the  vernier  is  -]'jdis, 
or  I  jV  of  a  divi-ion  on  tiie  hirnb.  If,  tlicn.  th.e  ver- 
nier be  amoved  till  1  Coincides  wiih  i,  0  -.viil  be  -jV/tli 
of  a  diivi-ion  of  the  liir.b  bcv'ond  o;  if  2  coincide 
with  :,  0  will  be  'i"-tiis  beyond  o;  and  so  on.  Hence 
to  read  the  vernier  in  any  position,  note  first  t!ie 
division  next  to  o,  and  bclnnd  it  on  the  limb.  This 
is  th.e  u:tc::rai  number  of  divisions  to  be  read.  'For 
the  fractional  part,  see  which  division  of  the  vernier 
is  in  a  line  with  one  on  the  limb;  if  it  be  the  4th 
(as  m  the  fi^'are',  that  indicates  an  addition  to  the 
read!^;:  of  -,.',, d^^  of  a  divi-ion  of  tl'ic  linib;  and  so  on. 
Thus,  if  die  iijiire  represent  a  barometer  scale  divided 

into  inches  and  tenths,  the  readin,::  is  30'34,  the  zero 

line  of  tile  Vernier  being  adjusted  lo  the  le\el  of  the  mercury. 

375.  If  the  limlj  of  a  se.xtant  be  di\ided,  as  it  usually  is,  to  third- 
parts  of  a  degree,  and  the  vernier  be  formed  !;y  dixiding  twenty-one 
of  these  into  twenty  equal  parts,  the  instrument  can  be  read  to 
twentieths  of  divisions  on  the  limb,  that  is,  to  minutes  of  arc. 

If  no  line  on  tlie  vernier  coincide  with  one  on  the  hmb,  tlien  since 
the  divisions  of  the  former  are  the  lon.:er  th.ere  v.-ill  be  one  of  the 
latter  included  between  the  two  lines  of  tiie  \ernier,  and  it  is  usual 
in  j/ractice  to  take  the  mean  of  the  readings  wliich  would  be  jjiven 
by  a  coincidence  of  either  pair  of  bounding;  lines. 

376.  In  the  above  sketch  and  description,  the  ntimbers  on  tlie 
scr,!,,  r.nd  \-cin;er  liave  been  sup[iOH:d  to  run  c/A-i.'/t'  v.-ays.  Tiiis 
i-  geTTcrahy  the  c:-;se  v.r'.i  British  instruments.  In  some  fjreign  'jnes 
tile  (ii'vision^.  ri.ri  in  the  same  direction  o\\  vernier  and  l.nib,  and  m 
that  case  it  js  ea^y  to  see  that  to  read  to  tenths  of  a  scale  dn.ision  we 
nrav.  liave  ten  di\a-;ons  of  the  vernier  equal  tq  nine  of  tlie  sc\die. 

In  genera!  to  read  to  the  iiih  part  of  a  scale  division,  ;;  dn.isions  of 
the  vernier  must  ecjual  ;/  -  i  or  ,v  -  r  divisi'.^.ns  on  the  limb,  according 
as  these  run  m  opp'^site  or  similar  directions.' 

377.  i'lie  princijde  of  the  Scrcu  has  been  already  noticed  (§  1 14). 
It  m.gv  be  Used  m  eitiier  of  two  ways,  i.e.  tiic  nut  may  be  fixed, 
and  t'.ie  screw  adv.jice  through  it,  or  the  i^crew  may  be  pre\'ented 
from  n.O'.dng  lo:i,^:tu;iinal!y  by  a  fi.xed  collar,  m  wli<ch  case  the  nut, 
if  prevented  !<;/  l.xed  guides  from  rotating,  will  move  m  the  directi  n 
of  the  comniun  axis.  The  advance  in  either  case  is  evidently  pr:-- 
porlional  to  the  angle  tluough  which  the  ijcrew  has  turned  abjut  itri 


130  PRELIMINARY. 

axis,  and  this  may  be  measured  by  means  of  a  divided  head  fixed 

perpendicularly  to  the  screw  at  one  end,  the  divisions  being  read  off 
by  a  pointer  or  vernier  attached  to  the  frame  of  the  instrument.  iThe 
nut  carries  with  it  either  a  tracing^oint  (as  in  the  dividing  engine)  or 
a  wire,  thread,  or  half  the  object-glass  of  a  telescope  (as  in  micro- 
meters), the  thread  or  wire,  or  the  play  of  the  tracing-point,  being 
at  right  angles  to  the  axis  of  the  screw. 

378.  Suppose  it  be  required  to  divide  a  line  into  any  number 
of  equal  parts.  The  line  is  placed  parallel  to  the  axis'  of  the  screw 
with  one  end  exactly  under  the  tracing-point,  or  under  the  fixed  wire 
of  a  microscope  earned  by  the  nut,  and  the  screw-head  is  read  off.  By 
turning  the  head,  the  tracing-point  or  microscope  wire  is  brought  to  the 
other  extremity  of  the  Hne;  and  the  number  of  turns  and  fractions  of 
a  turn  required  for  the  whole  line  is  thus  ascertained.  Dividing  this 
by  the  number  of  equal  parts  required,  we  find  at  oijce  the  number 
of  turns  and  fractional  parts  corresponding  to  one  of  the  required 
divisions,  and  by  giving  that  amount  of  rotation  to  the  screw  over  and 
over  again,  drawing  a  line  after  each  rotation,  the  required  division  is 
effected, 

379.  In  the  Micrometer,  the  movable  wire  carried  by  the  nut 
is  parallel  to  a  fixed  wire.  By  bringing  them  into  optical  contact 
the  zero  reading  of  the  head  is  known;  hence  when  another  reading 
has  been  obtained,  we  have  by  subtraction  the  number  of  turns 
corresponding  to  the  length  of  the  object  to  be  measured.  The 
absolute  value  of  a  turn  of  the  screw  is  determined  by  calculation 
from  the  number  of  threads  in  an  inch,  or  by  actually  applying  the 
micrometer  to  an  object  of  known  dimensions. 

380.  For  the  measurement  of  the  thickness  of  a  plate,  or  the  cur- 
vature of  a  lens,  the  Spheromeicr  is  used.  It  consists  of  a  screw  nut 
rigidly  fixed  in  the  middle  of  a  very  rigid  three-legged  table,  with  its 
axis  perpendicular  to  the  plane  of  the  three  feet  (or  finely  rounded 
ends  of  the  legs,)  and  an  accurately  cut  screw  working  in  this  nut. 
The  lower  extremity  of  the  screw  is  also  finely  rounded.  The  number 
of  turns,  whole  or  fractional,  of  the  screw,  is  read  off  by  a  divided 
head  and  a  pointer  fixed  to  the  stem.  Suppose  it  be  required  to 
measure  the  thickness  of  a  plate  of  glass.  The  three  feet  of  the 
instrum.ent  are  placed  upon  a  nearly  enough  flat  surface  of  a  hard 
body,  and  the  screw  is  gradually  turned  until  its  point  touches  and 
presses  the  surface.  The  muscular  sense  of  touch  perceives  resistance  to 
the  turning  of  the  screw  when,  after  touching  the  hard  body,  it  presses 
on  it  with  a  force  somewhat  exceeding  the  weight  of  the  screw.  The 
first  effect  of  the  contact  is  a  diminution  of  resistance  to  the  turning, 
due  to  the  weight  of  the  screw  coming  to  be  borne  on  its  fine  pointed 
end  instead  of  on  the  thread  of  the  nut.  The  sudden  increase  of 
resistance  at  the  instant  when  the  screw  commences  to  bear  part  of 
the  weight  of  the  nut  finds  the  sense  prepared  to  perceive  it  with  re- 
markable delicacy  on  account  of  its  contrast  with  the  immediately 
preceding  diminution  of  resistance.     The  screw-head  is  now  read  off, 


MEASURES  AND  INSTRUMENTS  131 

and  the  screw  turned  backwards  until  room  is  left  for  the  insertion, 
beneath  its  point,  of  the  plate  whose  thickness  is  to  be  measured. 
The  screw  is  again  turned  until  increase  of  resistance  is  again  per- 
ceived; and  the  screw-head  is  again  read  off.  The  difference  of  the 
readings  of  the  head  is  equal  to  the  thickness  of  the  plate,  reckoned 
in  the  proper  unit  of  the  screw  and  the  division  of  its  head. 

381.  If  the  cun'ature  of  a  lens  is  to  be  measured,  the  instrument 
is  first  placed,  as  before,  on  a  plane  surface,  and  the  reading  for  the 
contact  is  taken.  The  same  operation  is  repeated  on  the  spherical 
surface.  The  difference  of  the  screw  readings  is  evidently  the 
greatest  thickness  of  the  glass  which  would  be  cut  off  by  a  plane 
passing  through  the  three  feet.  This  is  sufficient,  with  tlie  distance 
between  each  pair  of  feet,  to  enable  us  to  calculate  the  radius  of  the 
spherical  surface. 

In  fact  if  a  be  the  distance  between  each  pair  of  feet,  /  the  length 
of  screw  corresponding  to  the  difference  of  the  two  readings,  R  the 

2 

radius  of  the  spherical  surface;  we  have  at  once  2R  =  —.+  !,  or,  as  / 

is  generally  very  small  compared  with  c,  the  diameter  is,  very  ap- 

proximately,  — ,. 
3^ 

382.  The  Cathetoieter  is  used  for  the  accurate  determination  of 
differences  of  level — for  instance,  in  measuring  the  height  to  which  a 
fluid  rises  in  a  capillary  tube  above  the  exterior  free  surface.  It 
consists  of  a  long  divided  metallic  stem,  turning  round  an  axis  as 
nearly  as  may  be  parallel  to  its  length,  on  a  fixed  tripod  stand:  and, 
attached  to  the  stem,  a  spirit-level.  Upon  the  stem  slides  a  metallic 
piece  bearing  a  telescope  of  which  the  length  is  approximately  enough 
perpendicular  to  the  axis.  The  telescope  tube  is  as  nearly  as  may  be 
perpendicular  to  the  length  of  the  stem.  By  levelling. screws  in  two 
feet  of  the  tripod  the  bubble  of  the  spirit-level  is  brought  to  one 
position  of  its  glass  when  the  stem  is  turned  all  round  its  axis.  This 
secures  that  the  axis  is  vertical.  In  using  the  instrument  the 
telescope  is  directed  in  succession  to  the  two  objects  whose  difference 
of  level  is  to  be  found,  and  in  each  case  moved  (generally  by  a 
delicate  screw)  up  or  down  the  stem,  until  a  horizontal  wire  in  the 
focus  of  its  eye-piece  coincides  with  the  image  of  the  object.  The 
difference  of  readings  on  the  vertical  stem  (each  taken  generally  by 
aid  of  a  vernier  sliding  piece)  corresponding  to  the  two  positions  of 
the  telescope  gives  the  required  difference  of  level. 

383.  The  principle  of  the  Balance  is  generally  kno^m.  We  may 
note  here  a  few  of  the  precautions  adopted  in  the  best  balances  to 
guard  against  the  various  defects  to  which  the  instrument  is  liable ; 
and  the  chief  points  to  be  attended  to  in  its  construction  to  secure 
delicacy,  and  rapidity  of  weighing. 

The  balance-beam  should  be  very  stiff,  and  as  light  as  possible 
consistently  with  the  requisite   stiffness.      For  this  puq^ose  it  is 


132  PRELIMINARY. 

generalfy  formed  either  of  tubes,  or  of  a  sort  of  lattice-framework. 
To  avoid  friction,  the  axle  consists  of  a  knife-edge,  as  it  is  called ; 
that  is,  a  wedge  of  hard  steel,  which,  when  the  balance  is  in  use, 
rests  on  horizontal  plates  of  polished  agate.  A  similar  contrivance 
is  applied  in  very  delicate  balances  at  the  points  of  the  beam  from 
which  the  scale-pans  are  suspended.  When  not  in  use,  and  just- 
before  use,  the  beam  with  its  knife-edge  is  lifted  by  a  lever  arrange- 
ment from  the  agate  plates.  While  thus  secured  it  is  loaded  with 
weights  as  nearly  as  possible  equal  (this  can  be  attained  by  previous 
trial  with  a  coarser  instrument),  and  the  accurate  determmation  is 
then  readily  effected.  The  last  fraction  of  the  required  weight  is 
detemiined  by  a  rider,  a  very  small  weight,  generally  formed  of  wire, 
which  can  be  worked  (by  a  lever)  from  the  outside  of  the  glass  case 
in  which  the  balance  is  enclosed,  and  which  may  be  placed  in 
different  positions  upon  one  arm  of  the  beam.  This  arm  is  gra- 
duated to  tenths,  etc.,  and  thus  shows  at  once  the  value  of  the  rider  in 
any  case  as  depending  on  its  moment  or  leverage,  §  233. 

384.  Qualities  of  a  balance: 

1.  Stability. — For  stability  of  the  beam  alone  without  pans  and 
weights,  its  centre  of  gravity  must  be  below  its  bearing  knife-edge. 
For  stability  with  the  heaviest  weights  the  line  joining  the  points  at 
the  ends  of  the  beam  from  which  the  pans  are  hung  must  be  below 
the  knife-edge  bearing  the  whole. 

2.  Sensibility. — The  beam  .should  be  sensibly  deflected  from  a 
horizontal  position  by  the  smallest  difference  between  the  weights  in 
the  sc'ale-pans.  The  definite  m.easure  of  the  sensibility  is  the  angle 
through  which  the  beam  is  deflected  by  a  stated  difference  between 
the  loads  in  the  pans. 

3.  Quickness. — This  means  rapidity  of  oscillation,  and  consequently 
speed  in  the  performance  of  a  weighing.  It  depends  mainly  upon  the 
depth  of  the  centre  of  gravity  of  the  whole  below  the  knife-edge  and 
the  length  of  the  beam. 

In  our  Chapter  on  Statics  we  shall  give  the  investigation.  The 
sensibility  and  quickness  are  calculated  for  any  given  form  and 
dimensions  of  the  instrument,  in  §  592. 

A  fine  balance  should  turn  with  about  a  5oo,oocth  of  the  greatest 
load  which  can  safely  be  placed  in  either  pan. 

The  process  oi  Double  Weighing,  which  consists  in  counterpoismg  a 
mass  by  shot,  or  sand,  or  pieces  of  fine  wire,  and  then  substituting 
weights  for  it  in  the  same  pan  till  equilibrium  is  attained,  is  more 
laborious,  but  more  accurate,  than  single  weighing ;  as  it  eliminates 
all  6rrors  arising  from  unequal  length  of  the  arms,  etc. 

Correction  is  reciuired  for  the  weights  of  air  displaced  by  the  two 
bodies  weighed  against  one  another  whert  their  difference  is  too  large 
to  be  negligable. 

385.  In  the  Toysifln-balance  invented,  and  used"  with  great  effect, 
by  Coulomb,  a  force  is  measured  by  tI"Le  torsion  of  a  fibre  of  silk,  a 
glass  thread,  or  a  metallic  wire.     The  fibre  or  wire  .is  fi.xed  at  its 


MEASURES  AND  IXSTR  U ME  NTS.  1 3  3 

upper  end,  or  at  both  ends,  according  to  circum~tnnces.  In  genera! 
it  carries  a  very  light  horizontal  rod  or  ncjuie,  to  the  c\:reniities  of 
which  are  attached  the  body  on  which,  is  cxcrt-jd  I'.c  l':.rce  to  be 
mcisurei,!,  and  a  counterpoise.  The  upper  cxtren-Miy  of  tiic  tor.  ion 
fibre  is  fixed  to  an  index  passing  through  i;:e  ccn-.re  of  a  divided 
disc,  so  that  the  angle  through  which  that  cxtrenvity  moves  ii  directly 
measured.  If,  at  the  same  time,  the  angle  tiirough  wiiiel-i  tiiC  needle 
has  turned  be  measured,  or,  more  simpl)-,  if  th,e  im'.cx  be  a';'v.:.y5 
turned  till  the  needle  assumes  a  dinerent  posirion  d-.tcrminvd  by 
marks  or  sights  attached  to  the  case  of  the  instrument — we  have  the 
amount  of  torsion  of  the  fibre,  and  it  becomes  a  simple  statical  j,ro- 
blem  to  detcrn.ine  frem  the  latter  the  force  to  be  measLir-ju  ;  its  direc- 
tion, and  point  of  application,  and  the  dimensions  of  tlie  appaiatus. 
being  known.  The  force  of  torsion  as  dej^ending  on  tlie  angle  of 
torsion  was  found  by  Coulonib  to  follow  the  law  of  simple  proi-uriion 
up  to  the  limits  of  perfect  elasticity — as  miglit  have  been  exiiccted 
from  Hooke's  Lav.-  (see  Properties  of  AfalL-r),  and  it  only  re.i-raMis 
that  we  determine  the  amount  for  a  particular  angle  in  absolute 
measure.  This  determination  is,  in  general,  simple  enough  in  theory; 
"but  in  practice  requires  considerable  care  and  nicety.  The  torsion- 
balance,  however,  being  chiefly  used  for  cumparative,  not  ab.-.oiu:e, 
measure,  this  determination  is  often  unnecessary.  More  will  be  said 
about  it  wlien  we  come  to  its  aiiplication. 

386.  The  ordinary  spiral  sprmg-balances  used  for  roughly  com- 
paring either  small  or  large  weights  or  forces,  are,  properly  speaking, 
only  a  modified  form  of  torsion-balance',  as  they  act  almost  entirely 
by  the  tor.-,ion  of  the  wire,  and  not  by  longitudinal  extension  or  by 
flexure.  Spring-balances  we  believe  to  be  capable,  if  carefully  con- 
structed, of  rivalling  the  ordinary  balance  in  accuracy,  while,  for  some 
applications,  they  far  surpa-s  it  in  sensibilicy  and  converrience.  They 
measure  directly  force,  not  mass;  and  therefore  if  used  for  deter- 
mining masses  in  different  ])arts  of  the  earth,  a  corrction  mj.st  be 
apphed  for  the  varying  force  of  gravity.  The  correction  for  tem- 
perature must  not  be  overlooked.  These  corrections  may  be  avoided 
by  the  method  of  double  weighing. 

387.  Perhaps  the  most  delicate  of  all  instruments  for  the  measure- 
ment of  force  is  the  Pc?idu!um.  It  is  proved  m  Kinetics  (see  Div.  II.) 
that  for  any  pendulum,  whether  oscillating  about  a  mean  \ert;ral 
position  under  the  action  of  gravity,  or  in  a  horizontal  jTane,  under 
the  action  of  magnetic  force,  or  force  of  torsion,  the  s-juare  of  the 
number  of  small  oscillations  in  a  given  time  is  pro;jortional  to  the 
magnitude  of  the  force  under  which  these  oscillations  take  place. 

P'or  the  estimation  of  the  relative  amounts  of  gravity  at  ditterent 
(places,  this  is  by  far  the  most  perfect  instrument.  The  method  of 
coincidences  by  which  this  process  has  been  rendered  so  excessively 
delicate  will  be  described  later. 

'  Binet.     See  alio  J.' Thomson.     Cambridge  and  Dublin  Mc.'.h.  J  cum:.'.,  1S43. 


134  PRELIMINARY. 

In  fact,  the  kinetic  measure  of  force,  as  it  is  the  first  and 
most  truly  elementary,  is  also  far  the  most  easy  as  well  as  perfect 
method  in  many  practical  cases.  It  admits  of  an  easy  reduction  to 
gravitation  measure. 

388.  Weber  and  Gauss,  in  constructing  apparatus  for  observations 
of  terrestrial  magnetism,  endeavoured  so  to  modify  them  as  to  admit 
of  their  being  read  from  some  distance.  For  this  purpose  each  bar, 
made  at  that  time  too  ponderous,  carried  a  plain  mirror.  By  means 
of  a  scale,  seen  after  reflection  in  the  mirror  and  carefully  read  with 
a  telescope,  it  was  of  course  easy  to  compute  the  deviations  which  the 
mirror  had  experienced.  But,  for  many  reasons,  it  was  deemed  neces- 
sary that  the  deflections,  even  under  considerable  force,  should  be 
very  small.  With  this  view  the  Bifilar  suspension  was  introduced. 
The  bar-magnet  is  suspended  horizontally  by  two  vertical  wires  or 
fibres  of  equal  length  so  adjusted  as  to  share  its  weight  equally 
between  them.  When  the  bar  turns,  the  suspension-fibres  become 
inclined  to  the  vertical,  and  therefore  the  bar  must  rise.  Hence,  if 
we  neglect  the  torsion  of  the  fibres,  the  bifilar  actually  measures  a 
force  by  comparing  it  with  the  weight  of  the  suspended  magnet. 

Let  a  be  the  half  length  of  the  bar  between  the  points  of  attach- 
ment of  the  wires,  Q  the  angle  through  which  the  bar  has  been  turned 
(in  a  horizontal  plane)  from  its  position  of  equilibrium,  /  the  length 
of  one  of  the  wires. 

Then  if  <2  be  the  couple  tending  to  turn  the  bar,  and  W\\.%  weight, 

we  have 


which  gives  the  couple  in  terms  of  the  deflection  B, 

If  the  torsion  of  the  fibres  be  taken  into  account,  it  will  be 
sensibly  equal  to  Q  (since  the  greatest  inclination  to  the  vertical 
is  small),  and  therefore  the  couple  resulting  from  it  will  be  EQ, 
where  E  is  some  constant.  This  must  be  added  to  the  value  of  Q 
just  found  in  order  to  get  the  whole  deflecting  couple. 

389.  Ergometers  are  instruments  for  measuring  energy.  Whites 
friction  brake  me'asures  the  amount  of  work  actually  performed  in 

any  time-  by  an  engine  or  other  '  prime  mover,'  by  allowing  it  during 
the  time  of  trial  to  waste  all  its  work  on  friction.  Morin's  crgometer 
measures  work  without  wasting  any  of  it,  in  the  course  of  its  trans- 
mission from  the  prime  mover  to  machines  in  which  it  is  usefully 
employed.  It  consists  of  a  simple  arrangement  of  springs,  measur- 
ing at  every  instant  the  couple  with  v/hich  the  prime  mover  turns  the 
shaft  that  transmits  its  work,  and  an  integrating  machine  from  which 
the  work  done  by  this  couple  during  any  time  can  be  read  off. 

390.  White's  friction  brake  consists  of  a  lever  clamped  to  the 
shaft,  but  not  allowed  to  turn  with  it.  The  moment  of  the  force 
required  to  prevent   tiie  lever  from  going  round  with  the  shaft, 


AfEASURES  AND  LVSTJ^U-UENTS.  135 

multiplied  by  the  whole  angle  through  whicli  the  sliaft  turns,  measures- 
the  whole  work  done  against  the  Iriclion  of  tin.  clarnp.  The  same 
result  is  much  more  easily  obtained  by  v/r.if);iin:,'  a  rope  or  chain 
several  times  round  the  shaft,  or  round  a  cylinder  or  druin  carried 
round  by  the  shaft,  and  applying  measured  forces  to  its  tv.o  ends 
in  proper  directions  to  keep  it  nearly  steady  while  the  shaft  turns 
round  without  it.  The  difference  of  the  moments  of  these  two  forces 
round  the  axis,  multiplied  by  the  angle  through  v.-hich  tiie  sh:ifl  tun^-, 
measures  the  whole  work  spent  on  friction  against  the  rope.  If  we 
remove  all  other  resistance  to  the  shaft,  and  apply  the  propicr  amount 
of  force  at  each  end  of  the  rope  or  chain  (which  is  very  ea:?ily  done 
in  practice),  the  prime  mover  is  kept  running  at  the  proper  spcerl 
for  the  test,  and  having  its  whole  work  thus  wasted  for  the  time  and 
measured. 


DIVISION   11. 

ABSTRACT     DYNAMICS. 

CHAPTER  v.— INTRODUCTORY. 

391.  Until  've  know  thoroughly  the  nature  of  matter  and  the 
forces  which  produce  its  motions,  it  will  be  utterly  impossible  to 
submit  to  mathematical  reasoning  the  exact  conditions  of  any  phy- 
sical question.  .  It  has  been  long  understood,  however,  that  an  ap- 
proximate solution  of  almost  any  problem  in  the  ordinary  branches 
of  Natural  Philosophy  may  be  easily  obtained  by  a  species  of  ab- 
stractmi,  or  rather  limitation  of  tJie  data,  such  as  enables  us  easily 
to  solve  the  modified  form  of  the  question,  while  we  are  well  assured 
that  the  circumstances  (so  modified)  affect  the  result  only  in  a  super- 
ficial manner. 

392.  Take,  for  instance,  the  very  simple  case  of  a  crowbar  em- 
ployed to  move  a  heavy  mass.  The  accurate  mathematical  investi- 
gation of  the  action  would  involve  the  simultaneous  treatment  of  the 
motions  of  every  part  of  bar,  fulcrum,  and  mass  raised;  and  from  our 
almost  complete  ignorance  of  the  nature  of  matter  and  molecular 
forces,  it  is  clear  that  such  a  treatment  of  the  problem  is  impossible. 

It  is  a  result  of  observation  that  the  particles  of  the  bar,  fulcrum, 
and  mass,  separately,  retain  throughout  the  process  nearly  the  same 
relative  positions.  Hence  the  idea  of  solving,  instead  of  the  above 
impossible  question,  another,  in  reality  quite  different,  but,  while 
infinitely  simpler,  obviously  leading  to  nearly  the  same  results  as  the 
former. 

393.  The  new  form  is  given  at  once  by  the  experimental  result 
of  the  trial.  Imagine  the  masses  involved  to  he  per/at ly  rigid  (i.Q. 
incapable  of  changing  their  forms  or  dimensions),  anrl  the  infinite 
multiplicity  of  the  forces,  really  acting,  may  be  left  out  of  consi- 
deration; so  that  the  rnaLhcmatical  investigation  deals  with  a  finite 
(and  generally  small)  number  of  forces  instead  of  a  practically  infinite 
number.     Our  warrant  for  such  a  substitution  is  established  thus. 


ABSTRACT  DYNAMICS.  lyi 

394.  The  only  effects  of  the  intermolecular  forces  would  be  ex- 
hibited in  molecular  alterations  of  the  form  or  volume  of  the  masses 
involved.  But  as  these  (practically)  remain  almost  unchanged,  the 
forces  which  produce,  or  tend  to  produce,  changes  in  them  may  be 
left  out  of  consideration.  Thus  we  are  enabled  to  investigate  the 
action  of  machinery'  by  supposing  it  to  consist  of  separate  portions 
whose  forms  and  dimensions  are  unalterable. 

395.  If  we  go  a  little  farther  into  the  question,  we  find  that  the 
lever  bends,  some  parts  of  it  are  extended  and  others  compressed. 
This  would  lead  us  into  a  very  serious  and  difficult  inquiry  if  v.-e  had 
to  take  account  of  the  whole  circumstances.  But  (by  experience)  we 
find  that  a  sufficiently  accurate  solution  of  this  more  formidable  case 
of  the  problem  may  be  obtained  by  supposing  (what  can  ?irL'er  be 
realized  in  practice)  the  mass  to  be  homogeneous,  and  the  forces 
consequent  on  a  dilatation,  compression,  or  distortion,  to  be  propor- 
tional in  magnitude,  and  opposed  in  direction,  to  these  deformations 
respectively.  By  this  farther  assumption,  close  approximations  may 
be  made  to  the  vibrations  of  rods,  plates,  etc.,  as  well  as  to  the  statical 
effects  of  springs,  etc. 

396.  W'e  may  pursue  the  process  farther.  Compression,  in  general, 
develops  heat,  and  extension,  cold.  These  alier  sensibly  the  elas- 
ticity of  a  body.  By  introducing  such  considerations,  we  reach, 
without  great  difficulty,  what  may  be  called  a  third  approximation 
to  the  solution  of  the  physical  problem  considered. 

397.  We  might  next  introduce  the  conduction  of  the  heat,  so 
produced,  from  point  to  point  of  the  solid,  with  its  accompanying 
modifications  of  elasticity,  and  so  on  ;  and  we  might  then  consider 
the  production  of  thermo-electric  currents,  which  (as  we  shall  see) 
are  always  developed  by  unequal  heating  in  a  mass  if  it  be  not  per- 
fectly homogeneous.  Enough,  however,  has  been  said  to  show,_/?>j-/, 
our  utter  ignorance  as  to  the  true  and  complete  solution  of  any 
pjhysical  (juestion  by  the  only  perfect  method,  that  of  the  consideration 
of  the  circumstances  which  affect  the  motion  of  every  portion,  sepa- 
rately, of  each  body  concerned;  and,  second,  the  practically  sufficient 
manner  in  which  practical  questions  maybe  attacked  by  limiting  their 
generality,  the  /imitations  introduced  being  themselves  deduced  from  ex- 
perience, and  being  therefore   Nature's  own    solution    (to   a   less   or 

greater    degree    of  accuracy)    of  the    infinite   additional  number  of 
equations  by  which  we  should  otherwise  have  been  encumbered. 

398.  To  take  another  case :  in  tlie  consideration  of  the  propa- 
gation of  waves  on  the  surface  of  a  tluid,  it  is  impossible,  not  only 
on  account  of  mathematical  difficulties,  but  on  account  of  our  igno 
ranee  of  what  matter  is,  and  what  forces  its  particles  exert  on  each 
other,  to  form  the  equations  which  would  give  us  the  separate  motion 
of  each.  Our  first  approximation  to  a  solution,  and  one  suftieient 
for  most  practical  purposes,  is  derived  from  the  consideration  of  the 


138  INTRODUCTORY. 

motion  of  a  homogeneous,  incompressible,  and  perfectly  plastic  mass; 
a  hypothetical  substance  which,  of  course,  nowhere  exists  in  nature. 

399.  Looking  a  little  more  closely,  we  find  that  the  actual  motion 
'differs  considerably  from  that  given  by  the  analytical  solution  of  the 
restricted  problem,  and  we  introduce  farther  considerations,  such  as 
the  compressibility  of  fluids,  their  internal  friction,  the  heat  generated 
by  the  latter,  and  its  effects  in  dilating  the  mass,  etc.  etc.  By  such 
successive  corrections  we  attain,  at  length,  to  a  mathematical  result 
which  (at  all  events  in  the  present  state  of  experimental  science) 
agrees,  within  the  limits  of  experimental  error,  with  observation. 

400.  It  would  be  easy  to  give  many  more  instances  substantiating 
what  has  just  been  advanced,  but  it  seems  scarcely  necessary  to  do 
so.  We  may  therefore  at  once  say  that  there  is  no  question  in 
physical  science  which  can  be  cotnpletcly  and  accurately  investigated 
by  mathematical  reasoning  (in  which,  be  it  carefully  remembered, 
it  is  not  necessary  that  symbols  should  be  introduced),  but  that  there 
are  different  degrees  of  approximation,  involving  assumptions  more 
and  more  nearly  coincident  with,  Observation,  which  may  be  arrived 
at  in  the  solution  of  any  particular  question. 

401.  The  object  of  the  present  division  of  this  work  is  to  deal  with  the 
first  and  second  of  these  approximations.     In  it  we  shall  suppose  all 

solids  either  rigid,  i.e.  unchangeable  in  form  and  volume,  or  elastic; 
but  in  the  latter  case,  we  shall  assume  the  law,  connecting  a  com- 
pression or  a  distortion  with  the  force  which  causes  it,  to  have  a 
particular  form  deduced  from  experiment.  And  we  shall  also  leave 
out  of  consideration  the  thermal  or  electric  effects  which  compression 
or  distortion  generally  produce.  We  shall  also  suppose  fluids,  whether 
liquids  or  gases,  to  be  either  incompressible  or  compressible  ac- 
cording to  certain  known  laws ;  and  we  shall  omit  considerations 
of  fluid  friction,  although  we  admit  the  consideration  of  friction 
between  solids.  Fluids  will  therefore  be  supposed  perfect,  i.e.  such 
that  any  particle  may  be  moved  amongst  the  others  by  the  slightest 
force. 

402.  AMien  we  come  to  Properties  of  Matter' and  the  Physical 
Forces,  we  shall  give  in  detail,  as  far  as  they  are  yet  known,  the 
modifications  which  farther  approximations  have  introduced  into  the 
previous  results. 

403.  The  laws  of  friction  between  solids  were  very  ably  investi- 
gated by  Coulomb;  and,  as  we  shall  require  them  in  the  succeeding 
chapters,  wc  give  a  brief  summary  of  them  here^  reserving  the  more' 
careful  scrutiny  of  experimental  results  to  our  chapter  on  Properties 
of  Matter. 

404.  To  produce  sliding  of  one  solid  body  on  another,  the  sur- 
faces in  contact  being  pkvnc,  recjuires  a  tangential  force  which 
depends, — (i)  upon  the  nature  of  the  botlies;  (2)  upon  their  polish, 
or  the  species  and  quantity  of  lubricant  which  may  have  been  applied; 


ABSTRACT  DYNAMICS.  139 

(3)  upon  the  normal  pressure  between  them,  to  which  it  is  in  general 
directly  proportional;  (4)  upon  the  length  of  time  during  which  they 
have  been  suffered  to  remain  in  contact. 

It  does  not  (except  in  extreme  cases  where  scratching  or  abrasion 
takes  place)  depend  sensibly  upon  the  area  of  the  surfaces  in  contact. 
This,  which  is  called  Statical  Friction,  is  thus  capable  of  opposing  a 
tangential  resistance  to  motion  which  may  be  of  any  requisite  amount 
up  to  /xi?;  where  R  is  the  whole  normal  pressure  between  the  bodies; 
and  /x  (which  depends  mainly  upon  the  nature  of  the  surfaces  in 
contact)  is  the  co-efficient  of  Statical  Friction.  This  co-efficient  varies 
greatly  with  the  circumstances,  being  in  some  cases  as  low  as  o"o3,  in 
others  as  high  as  o-8o.  Later  we  shall  give  a  table  of  its  values. 
Where  the  applied  forces  are  insufficient  to  produce  motion,  the 
whole  amount  of  statical  friction  is  not  called  into  play;  its  amount 
then  just  reaches  what  is  sufficient  to  equilibrate  the  other  forces,  and 
its  direction  is  the  opposite  of  that  in  which  their  resultant  tends  to 
produce  motion.  When  the  statical  friction  has  been  overcome,  and 
sliding  is  produced,  experiment  shows  that  a  force  of  friction  con- 
tinues to  act,  opposing  the  motion,  sensibly  proportional  to  the 
normal  pressure,  and  independent  of  the  velocity.  But  for  the  same 
two  bodies  the  co-effiicieni  of  Kinetic  Friction  is  less  than  that  of  Sta- 
tical Friction,  and  is  approximately  the  same  whatever  be  the  rate  of 
motion. 

405.  When  among  the  forces  acting  in  any  case  of  equilibrium, 
there  are  frictions  of  solids  on  solids,  the  circumstances  would  not 
be  altered  by  doing  away  with  all  friction,  and  replacing  its  forces  by 
forces  of  mutual  action  supposed  to  remain  unchanged  by  any  in- 
finitely small  relative  motions  of  the  parts  between  which  they  act. 
By  this  artifice  all  such  cases  may  be  brought  under  the  general 
principle  of  Lagrange  (§  254). 

406.  In  the  following  chapters  on  Abstract  Dynamics  we  will 
confine  ourselves  chiefly  to  such  portions  of  this  extensive  subject 
as  are  likely  to  be  useful  to  us  in  the  rest  of  tlie  work, 


CHAPTER-  VI. 

STATICS  OF  A  PARTICLE.-ATTRACTION. 


407.  We  naturally  divide  Statics  into  two  parts — the  equilibrium 
of  a  Particle,  and  that  of  a  rigid  or  elastic  Body  or  System  of  Particles 
whether  solid  or  fluid.  The  second  law  of  motion  suffices  for  one 
part — for  the  other,  the  third,  and  its  consequences  pointed  out  by 
Newton,  are  necessary.  In  the  succeeding  sections  we  shall  dispose 
of  the  first  of  these  parts,  and  the  rest  of  this  chapter  will  be  devoted 
to  a  digression  on  the  important  subject  of  Attraction. 

408.  By  §  2  2  1,  forces  acting  at  the  same  point,  or  on  the  same 
material  particle,  are  to  be  cornpounded  by  the  same  laws  as  velo- 
cities. Therefore  the  sum  of  their  resolved  parts  in  any  direction 
must  vanish  if  there  is  equiUbrium;  whence  the  necessary  and  suffi- 
cient conditions. 

They  follow  also  directly  from  Newton's  statement  with  regard  to 
work,  if  we  suppose  the  particle  to  have  any  velocity,  constant  in 
direction  and  magnitude  (and  §  211,  this  is  the  most  general  sup- 
position we  can  make,  since  absolute  rest  has  for  us  no  meaning). 
For  the  work  done  in  any  time  is  the  product  of  the  displacement 
during  that-time  into  the  algebraic  sum  of  the  effective  components 
of  the  applied  forces,  and  there  is  no  change  of  kinetic  energy. 
Hence  this  sum  must  vanish  for  every  direction.  Practically,  as  any 
displacement  may  be  resolved  into  three,  in  any  three  directions  not 
cojjlanar,  the  vanishing  of  the  work  for  any  one  such  set  of  three, 
suffices  for  the  criterion.  But,  in  general,  it  is  convenient  to  assume 
them  in  directions  at  right  angles  to  each  other. 

Hence,  for  the  equilibrium  of  a  material  particle,  it  is  necessary,  and 
sufficient,  that  the  (algebraic)  sums  of  the  applied  forces,  resolved  in 
any  one  set  of  three  rectangular  directions,  should  vanish. 

409.  We  proceed  to  give  a  detailed  exposition  of  the  results 
which  follow  from  the  first  clause  of  §  408,  For  three  forces  only  we 
have  the  following  statement. 

The  resultant  of  two  forces,  acting  on  a  material  point,  is  repre^ 


STATICS   OF  A   PARTICLE.— ATTRACTION.      141 


sented  in  direction  and  magnitude  by  the  diagonal,  through  that 
point,  of  the  parallelogram  described  upon  lines  representing  tbo 
forces. 

410.  Parallelogram  of  forces  stated  symmetrically  as  to  the   three 
forces  concerned,  usually  called  the   Triangle  of  Forces.     If  the  lines 

representing  three  forces  acting  on  a  material  point  be  equal  and 
parallel  to  the  sides  of  a  triangle,  and  in  directions  similar  to  those 
of  the  three  sides  when  taken  in  order  round  the  triangle,  the  three 
forces  are  in  equilibrium. 

Let  GEF  be  a.  triangle,  and 
let  MA.,  MB.,  MC,  be  respectively 
equal  and  parallel  to  the  three 
sides  EF,  FG,  GE  of  this  trian- 
gle, and  in  directions  similar  to 
the  consecutive  directions  of 
these  sides  in  order.  The  point 
M  \%  in  equilibrium. 

411.  \True  Triangle  of  Forces.  Let  three  forces  act  in  consecu- 
tive directions  round  a  triangle,  DEE.,  and  be 
represented  respectively  by  its  sides  :  they  are 
not  in  equilibrium,  but  are  equivalent  to  a 
couple.  To  prove  this,  through  D  draw  DH, 
equal  and  parallel  to  EF.,  and  in  it  introduce 
a  pair  of  balancing  forces,  each  equal  to  EF. 
Of  the  five  forces,  three,  DE,  DH  and  FD., 
are  in  equilibrium,  and  may  be  removed ; 
and  there  are  then  left  two  forces,  EF  and  HD,  equal,  parallel,  and 
in  dissimilar  directions,  which  constitute  a  couple.] 

412.  To  find  the  resultant  of  any  number  of  forces  in  lines  through. 
one  point,  not  necessarily  in  one  plane — 

Let  MA^,  MA^,  MA^,  MA^  repre- 
sent four  forces  acting  on  M.,  in  one 
plane;  required  their  resultant. 

Find  by  the  parallelogram  of  forces, 
the  resultant  of  two  of  the  forces,  MA.^ 
and  MA^.  It  will  be  represented  by 
MD'.  Then  similarly,  find  MD",  the 
resultant  of  MD'  (the  first  subsidiary 
resultant),  and  MA^,  the  third  force. 
Lastly,  find  MD"\  the  resultant  of 
Afiy  and  MA^.  MD"  represents  the 
resultant  of  the  given  forces. 

Thus,  by  successive  a^  ''cations  of  the  fundamental  propc^ition,. 
the  resultant  of  any  nup^u,,.  of  forces  in  lines  through  one  point  can 
be  found. 

413.  In  executing  this  construction,  it  is  not  neccssar,-  to  describe 


142 


ABSTRACT  DYNAMICS. 


the  successive  paraHelograms,  or  even  to  draw  their  diagonals.     It  is 

enough  to  draw  through  the  given  poinf 
a  hne  equal  and  parallel  to  the  repre* 
sentative  of  any  one  of  the  forces; 
through  the  point  thus  arrived  at,  to 
draw  a  line  equal  and  parallel  to  the 
representative  of  another  of  the  forces^ 
and  so  on  till  all  the  forces  have  been 
taken  into  account.  In  this  v/ay  we  get 
such  a  diagram  as  the  annexed. 

The  several  given  forces  may  be  taken 
in  any  order,  in  the  construction  just 
described.  The  resultant  arrived  at  is 
necessarily  the  same,  whatever  be  the  order  in  which  we  choose  to 
take  them,  as  we  may  easily  verify  by  elementary  geometry. 

In  the  fig.  the  order  is  MA^,  MA^f 
MA„  MA^,  MA^. 

414.  If,  by  drawing  lines  equal  and 
parallel  to  the  representatives  of  the  forces, 
a  closed  figure  is  got,  that  is,  if  the  line 
last  drawn  leads  us  back  to  the  point 
from  which  we  started,  the  forces  are  in 
equilibrium.  If,  on  the  other  hand,  the 
figure  is  not  closed  (§  413),  the  resultant 
is  obtained  by  drawing  a  Hne  from  the 
starting-point  to  the  point  finally  reached; 
{from  M  to  JD)'.  and  a  force  represented  by  DM  will  equilibrate  the 
system. 

415.  Hence,  in  general,  a  set  of  forces  represented  by  lines  equal 
and  parallel  to  the  sides  of  a  complete  polygon,  are  in  equilibrium, 
provided  they  act  in  Hues  through  one  point,  in  directions  similar  to 
the  directions  followed  in  going  round  the  polygon  in  one  way. 

416.  Polygon  of  Forces.  The  -construction  we  have  just  con- 
sidered, is  sometimes  called  the  polygon  of  forces ;  but  the  true 
polygon  of  forces,,  as  we  shall  call  it,  is  something  quite  different. 
In  it  the  forces  are  actually  along  the  sides  of  a  polygon,  and  repre- 
sented by  them  in  magnitude..  Such  a  system  must  clearly  have  a 
turning  tendency,  and  it  may  be  demonstrated  to  be  reducible  to  one 
couple. 

417.  In  the  preceding  sections  we  have  explained  the  principle 
involved  in  finding  the  resultant  of  any  number  of  forces.  We  have 
now  to  exhibit  a  method,  more  easy  than  the  parallelogram  of  forces 
affords,  for  working  it  out  in  actual  cases,  and  especially  for  obtaining 
a  convenient  specification  of  the  resultant.  The  instrument  employed 
for  this  purpose  is  Trigonometry. 

418.  A  distinction  may  first  be  pointed  out  between  two  classes 
of  problems,  direct  and  inverse.  Direct  problems  are  those  in  which 
the  resultant  of  forces  is  to  be  found;  inverse,  those  in  which  com- 


STATICS  OF  A  PARTICLE.—ATTRACTION.      143 

ponents  of  a  force  are  to  be  found  The  former  class  is  fixed  and 
determinate ;  the  latter  is  quite  indefinite,  without  limitations  to  be 
stated  for  each  problem.  A  system  of  forces  can  produce  only  one 
effect;  but  an  infinite  number  of  systems  can  be  obtained,  which 
shall  produce  the  same  efi'ect  as  one  force.  The  problem,  therefore, 
of  finding  components  must  be,  in  some  way  or  other,  limited.  This 
may  be  done  by  giving  the  lines  along  which  the  components  are  to 
act.  To  find  the  components  of  a  given  force,  in  any  three  given 
directions,  is,  in  general,  as  we  shall  see,  a  perfectly  determinate 
problem. 

Finding  resultants  is  called  Composition  of  Forces. 

Finding  components  is  called  Resolution  of  Forces. 

419.  Composition  of  Forces. 

Required  in  position  and  magnitude  the  resultant  of  two  given 
forces  acting  in  giving  lines  on  a  material  point. 

Let  AfA,  ATB  represent  two  forces, 
F  and  <2,  acting  on  a  material  point  M. 
Let  the  angle  BMA  be  denoted  by  t, 
Required  the  magnitude  of  the  resultant, 
and  its  inclination  to  the  line  of  either 
force.  3/  i^ 

Let  R  denote  the  magnitude  of  the  resultant;  let  a  denote  the 
angle  DMA,  at  which  its  line  MD  is  inclined  to  MA,  the  line  of  the 
first  force  P;  and  let  /?  denote  the  angle  DMB,  at  \\hich  it  is  inclined 
to  MB,  the  direction  of  the  force  Q. 

Given  Z',  Q,  and  i:  required  R,  and  a  or  /^. 
We  have 

Miy  =  MA^  +  MB'  -  2MA.MB  X  cos  MAD. 

Hence,  according  to  our  present  notation, 

R' =  F' +  Q'- 2 PQ  cos  {1^0' -i), 
or  «        R'^P'+Q'  +  2PQC0SU 

Hence  R  ={P'+Q'  +  2PQ  cos  i)K  (i) 

To  determine  a  and  /3  after  the  resultant  has  been  found ;  we  have 

sm  DMA  =  -^^2)  sin  ^^^F> 

or  sma  =  -|smi,  (2) 

K 

and  similarly, 

p 

sm  /3  =  • ,-  sin  i.  (3) 

420.  These  formulae  are  useful  for  many  applications ;  but  tliey 
have  the  inconvenience  that  there  may  be  nmbi,i^uity  as  to  the  an-;!e, 
whether.it  is  to  be  acute  or  obtuse,  which  is  to  be  taken  vrhcn  eitiicr 
sin  G  or  sin  /?  has  been  calculated.  If  i  is  acute,  both  a  an-:^  P.  are 
acute,  and  there  is  no  ambiguity.     If  i  is  obtuse,  one  of  the   two 


144  ABSTRACT  DYNAMICS. 

angles,  a,  j3,  might  be  either  acute  or  obtuse;  but  as  they  cannot  be 
both  obtuse,  the  smaller  of  the  two  must,  necessarily,  be  acute.  If,, 
therefore,  we  take  the  formula  for  sin  a,  or  for  sin  /?,  according  as  the 
force  P,  or  the  force  ^,  is  the  greater,  we  do  away  with  all  ambiguity, 
-and  have  merely  to  take  the  value  of  the  angle  shown  in  the  table  of 
sines.  And  by  subtracting  the  value  thus  found,  from  the  given 
value  of  t,  we  find  the  value,  whether  _acute  or  obtuse,  of  the,  other  of 
the  two  angles,  a,  /?. 

421.  To  determine  a  and  /3  otherwise.  After  the  magnitude  of  the 
resultant  has  been  found,  we  know  the  three  sides»  MA^.  AD^  MD^ 
of  the  triangle  DMA,  then  we  have 

^,...     Mjy^MA'^AD^ 
CO,  DMA  ^        ./,/Z^.j/4    ■   - 

or                                cosa=      2RF-'*  ^"^^ 

and  Similarly,  cos/3  = ^^ — ,  (5) 

by  successive  applications  of  the  elementary  trigonometrical  formula 
used  above  for  finding  MD.  Again,  using  this  last-mentioned  for- 
mula for  MD^  or  i?*  in  the  numerators  of  (4)  and  (5),  and  reducing, 

1                                           -^+  Qcos  c  ,,, 

.we  have  cosa= ~ ,  (6) 

o     Q  +  wPcos  c  .  ^ 

formulae  which  are  convenient  in  many  cases.  There  is  no  am- 
biguity in  the  determination  of  either  a  or  ^  by  any  of  the  four 
equations  (4),  (5),  (6),  {7). 

Remark. — Either  sign  (-t-  or  -)  might  be  given  to  the  radical 
in  (i),  and  the  true  hne  of  action  and  ihe  direction  of  the  force  in  it 
would  be  determined  without  ambiguity  by  substituting  in  (2)  and- 
(3)  the  value  of  R.  with  either  sign  prefixed.  Since,  however,  there 
can  be  no  doubt  as  to  the  direction  of  the  force  indicated,  it  will  be 
generally  convenient  to  give  the  positive  sign  to  the  value  of  R.  But 
in  special  cases,  the  negative  sign,  which  with  the  proper  interpre- 
tation of  the  formulae  will  lead  to  the  same  result  as  the  positive, 
will  be  employed. 

422.  Another  method  of  treating  the  general  problem,  which  is 
useful  in  many  cases,  is  this : 

Let 

1(^+0  =  ^ 

which  implies  that 

P=F+G, 
Q  =  P~G. 


STATICS   OF  A   PARTICLE.— ATTRACTION'.      145 


/^and  G^will  be  both  positive  \{  P>Q.     Hence,  instead  of  the  tvo 

given  forces,  P  and  Q,  we  may  suppose  that  we  have  on  the  point  IvC 

four  forces; — two,  each  equal  to  F,  acting  in 

the  same  directions,  MK,  ML.,  as  the  given 

forces,  and   two  others,  each  equal  to  G,  of 

which  one  acts  in  the  same  direction,  MK,  as 

P.,  and  the  other  in  J/Z',  the  direction  opposite 

to    Q.     Now  the  resultant  of  the  two   equal 

forces,  F,  bisects   the  angle  between   them, 

KML;.    and   by  the   investigation   of  §  423 

below,  its  magnitude  is  found  to  be  2Fco^  |  c. 

Again,  the  resultant  of  the  two  equal  forces, 

G,  is  similarly  seen  to  bisect  the  angle,  KML' y 

between  the  line  of  the  given  force,  /',  ana 

the  continuation  through  M  of  the  line  of  the  given  force,  Q\  and  to 

be  equal  to  26^  sin  ^  (,  since  the  angle  KML  is  the  supplement  of  u 

Thus,  instead  of  the  two  given  forces  in  lines  inclined  to  one  another 

'at  the  angle  t,  which  may  be  either  an  acute,  an  obtuse,  or  a  right 

angle,  we  have  two  forces,  zFcos  ^  i  and  2  6^  sin  i  i,  acting  in  lines, 

MS,  MT,  which  bisect  the  angles  LMK  and  KML'.,  and  therefore 

are  at  right  angles  to  one  another.     Now,  according  to  §  429  below, 

we  find  the  resultant'  of  these  two  forces  by  means  of  the  following 

formulae:— 

2(7  sin  At 

tan  SMD  =  —jr, r— , 

2r  cos  h  I 

R  -  2i^cos  \  I  sec  SMD, 

"p-Q 

P+Q 

R  =  {P+  Qi  cos  Usee  (hi- a) 

=  (P+(2)co3|(a  +  /3)secH^-/3)'  (9) 

These  formulae  might  have  been  derived  from  the  standard  formulae 
for  the  solution  of  a  plane  triangle  when  two  sides  (P  and  Q),  and' 
the  contained  angle  (tt-  t)  are  given. 

423.     We  shall  now  investigate  some  cases  of  the  general  formulae. 
Case  I.     Let  the  forces  be  equal,  that  is,  let  Q  =  P  in  the  preceding 
formulae. 

Then,  by  (i),     R' =  2P'+ 2P' cos  l=2P\i  +  cost) 

r^^/^'cosHr 
Hence  R  =  zi'cos  J  i, 

an   important  result  which   might,  of  course,  have  been  obtained 
directly  from  tlie  proper  geometrical  construction  in  this  case.     Also 

by  (2), 

*  In  the  diagram  the  tlixectioa  of  the  balancing  force  is  sho-vMi  by  tlie  arxow* 
tead  in  the  line  £>Af, 


and 


and 


tan  (I  t  -  a) : 


,  tan  J  t, 


tS) 


146  ABSTRACT  DYNAMICS. 

O  sin  t        O  sin  t         .    , 
Sin  a  =  ^  „     =  -^ — -r-  =  sin  i  t, 
R  2  (2  cos  4  I  ^  ^ 

which  agrees  with  what  we  see  intuitively,  that  o.-^~\u 

424.     Ca^  II.    Let  P=  Q\  and  let  t=  120".    Then 
cos  ^  t  =  cos  60"  =  I,  and  (§  423)  R  =  P. 

The  resultant,  therefore,  of  two  equal  forces  inclined  at  an  angle 
of  120"  is  equal  to  each  of  them.  This  result  is  interesting,  because 
it  can  be  obtained  very  simply,  and  quite  independently  of  this 
investigation,  A  consideration  of  the  symmetry  of  the  circumstances 
will  show  that  if  three  equal  forces  in  one  plane  be  applied  to  a 
material  point  in  lines  dividing  the  space  around  it  into  three  equal 
angles,  they  must  be  in  equilibrium ;  which  is  perfectly  equivalent  tQ 
the  preceding  conclusioh. 

•425.     Case  III    Let  t  =  o" ;    cos  t  =  i ; 

then  i?  =  {i^+(2^+2i'(2A 

i?  =  P+(2. 

426.  Case IV.    Lett  =180";    cost=:-ij 
then  R^{P'^(X-2PQ^', 

R=:P-Q. 
This  is  also  one  of  the  cases  in  which  it  is  convenient  to  give  some* 
times  the  negative  sign,  sometimes  the  positive  to  the  expression  for 
the  resultant  force:  for  if  Q  be  greater  than  P,  the  preceding  expression 
will  be  negative,  and  the  interpretation  will  be  found  by  considering 
that  the  force  which  vanishes  when  P-.Qrh  in  the  direction  of  P 
when  P  is  the' greater,  and  in  the  contrary  direction,  or  in  that  of  Qj 
tvhen  P  is  .the  less  of  the  two  forces. 

427.  Case  V.  Forces  nearly  conspiring.  Let  the  angle  i  be  very 
small,  then  sintsst;*  cost«i. 

The  general  expressions  (§419)  therefore  become, 
R^P+Q, 
<2' 

''""^^pTq:' 

To  the  same  dt^ree  of  approximation 

Hence  a +  /?- ^^^ +  -^'_(<2 +  /')»_ ^ 

Hence  a  +  /3~  -^, -^  ~  ---^^  -  u 

^  The  signi=is  used  to  denote  approximate  equality. 


STATICS  OF  A  PARTICLE.—ATTRACTJON.      147 

This  shows  that  the  errors  in  the  values  of  a  and  /3  obtained  ap- 
proximately by  this  method  compensate;  one  being  as  much  above, 
as  the  other  is  below,  the  true  value. 

We  therefore  conclude  that  the  resultant  of  two  forces  very  nearly 
conspiring  is  approximately  equal  to  their  sum,  and  approximately 
divides  the  angle  between  them  into  parts  inversely  as  the  forces. 

When  the  angle  between  the  forces  is  infinitely  small,  they  may 
either  conspire  in  acting  on  one  point  in  one  line ;  or  they  may  act 
on  different  points  in  parallel  lines.  In  either  case  the  resultant  is 
precisely  equal  to  their  sum.  Actually  conspiring  forces  we  have 
already  considered;  parallel  forces  we  shall  conside;- more  particularly 
when  we  treat  of  the  equilibrium  of  a  rigid  body. .  We  may  briefly 
ejcamine  the  case  here  however.  Suppose  the  actual  points  of  appli- 
cation of  the  forces  to  be  A 
and  i?,  but  let  their  lines 
meet  in  a  point  M;  join 
AB,  and  let  MAB  be  an 
isosceles  triangle.  Let  this 
pomt  M  be  removed  grad- 
ually to  an  infinite  distance  in  the  direction  of  a  perpendicular,  C*J/, 
bisecting  the  line  AB.  The  resultant  will  still  divide  the  angle  in- 
versely as  the  forces:  and  as  the  circular  measure  of  the  angle  is  any 
arc  described  from  yl/as  centre  divided  by  the  radius,  every  such  arc 
will  be  divided  in  the  same  proportion.  Now,  if  M  be  infinitely 
distant,  that  is  if  the  lines  of  the  forces  be  parallel,  the  arc  will  become 
a  straight  line,  and  will  be  divided  into  parts  inversely  as  the  forces. 

In  actual  cases  of  forces  acting  on  a  point,  and  very  nearly  con- 
spiring, the  following  approximate  equations  show  how  nearly  the 
resultant  approaches  the  sum  of  the  forces  : — 


R^j[{P^Qf 


BQ> 


(12) 


^{B^Qy_ 
,_PQ  , 

that  is,  the  resultant  of  two  forces  very  nearly  conspiring  falls  short  of 
their  sum  by  the  square  of  the  angle  between  them  multiplied  into  a 
quarter  of  their  harmonic  mean'. 

428.      Gisc  VI.     Forces  nearly  opposed. 

1°.     Let  the  angle  t  be  very  obtuse,  and  the  two  forces  exactly  equal. 


^  The  Harmonic  Mean,  of  two  numbers  is  the  reciprocal  of  the  mean  of  liieir 
reciprocals.     Thus  the  harmonic  mean  of  P  and  Q  is  -^ — p, . 


148  4'BSTRACT  DYNAMICS. 

Let  t  =  TT  -  ^,  where  B  is  very  small, 

then  \*-  =  \'^-\Q, 

cos  J  t  =  sin  ^  ^, 

^=2i'sin^^, 

and  since  the  sine  oi  a  very  small  angle  is  equal  to  the  angle,  in 
circular  measure  R'^PQ. 

Hence  the  resultant  of  two  equal  very  nearly  Opposed  forces  is 
proportional  to  the  defalcation  from  direct  opposition:  being  ap- 
proximately equal  to  either  of  the  forces  multiplied  into  the  supple- 
ment of  the  angle  between  them. 

2°.  If  the  forces  are  neither  equal  nor  nearly  equal,  the  resultant 
will  be  approximately  equal  to  their  difference, 

We  have  as  before, 


Therefore 


cos  ts=:- 

■^'. 

R'^P'^Q' 

-  2PQ. 

P^P- 

Q, 

Q{^ 
Smass^ 

-0 
Q  ' 

•    /?     Pi"" 

-0 

bill  p —      „_ 

■Q  ' 

The  ambiguity  as  to  whether  the  acute  angle,  shown  in  the  table, 
or  its  supplement,  is  to  be  chosen  in  either  case,  may  be  removed  by 
considering  which  of  the  two  forces  is  the  greater. 

Thus,  as  we  suppose  P  to  be  greater  than  Q,  a  is  acute,  and  there^ 

fore 

and  /?  is  obtuse. 

Therefore 


sin 

(2  (^  - ') 

fi- 

=  7r  ■ 

P(^-i) 
P-Q 

/5-= 

Pi-    QlT 

P-Q' 

a  +  /3=: 

P-Q 

or 

We  find,  by  addition, 


and  conclude,  as  in  the  former  case,  that  the  errors  in  the  approximate 
values  of  a  and  (S  compensate,  one  being  as  much  above,  as  the  other 
is  below,  the  true  value. 

It  is  only  when  P  is  comparable  in  magnitude  with  P  and  Q,  that 
the  foregoing  solution  is  appHcable. 

But  if /^exceeds  Q,  or  if  (2  exceeds  P,  by  any  difterence  which  is 
considerable  in  comparison  with  either,  the  formulae  hold. 

Let  us  suppose  now  that,  while  P  remains  of  any  constant  mag- 
nitude, Q  is  made  to  increase  from  nothing,  gradually,  until  it  becomes 


STATICS  GF  A   PARTICLE.— ATTRACTION.      149 

first  equal  to,  and  then  greater  than  P,  the  angle  i  remaining  constant. 
The  angle  a  will  increase  very  slowly,  according  to  the  approximate 
formula  (10),  until  Q  becomes  nearly  equal  to  P.  Then  as  the  value 
of  <2  is  increased  until  it  becomes  greater  than  P,  the  value  of  a  will 
increase  very  rapidly  through  nearly  two  right  angles,  until  it  falls 
but  little  short  of  i,  when  its  supplement  will  be  approximately  ex- 
pressed by  the  formula  (10). 

In  this  transition,  from  Q<P  to  Q>P,  the  direction  and  magni- 
tude of  the  resultant  are  most  conveniendy  found  by  means  of  (§  422) 
the  last  of  the  three  general  methods  given  above  for  determining  the 
resultant  of  two  forces. 

Thus,  instead  of  the  two  given  forces  we  may  substitute  two 
forces  in  lines  bisecting  respectively  the  obtuse  angle  LATK,  or  t, 
and  the  acute' angle  KAIL  and  of  magnitudes  which  approximate 
to  \{P+Q)  (TT-t),  and  P-Q  respectively,  when  t  is  nearly  two 
right  angles. 

We  infer,  finally,  that,  however  nearly  P  and  Q  are  equal  to  one 
another,  the  approximate  formulae  of  §  428,  2"  hold,  provided  only 
J  (/'+  0 (;r -  t)  is  a  small  fraction  oiP-  Q. 

429,     Case  VII.     Lei  1  =  90°;    cos  1  =  0,    sint=i; 
then.  I^  =  {P'+Q:)K  (13) 

and  sin  a  =  -7]  1 

sm  /?  =  -5 

In  this  case,  /3  being  the  complemerit  of  a,  sin  ^8=;  cos  a. 

Hence  cos  a  =  — 

A 


Lastly,  since 

sm  a 
tan  a  = , 

COSa' 

we  deduce 

tan  a  =  -^ , 

and 

R  =  Psi2ca. 

(15) 

(16) 

Remark. — 'These  formulae  have  thus  been  derived  from  the  general 
expression  (§  419);  but  they  can  also  be  very  readily  got  from  a 
special  geometrical  construction,  corresponding  to  the  case  in  which 
the  lines  of  the  forces  are  at  right  angles  to  one  another,  the  prin- 
ciples to  be  used  being  (i)  the  parallelogram  of  forces;  (2)  Euclid  L, 
XLVIL;  and  (3)  the  trigonometrical  definitions  of  sine,  cosine,  and 
tangent. 

430.  This  case  is  of  importance,  for  it  affords  us  the  formulae  for 
rectangular  resolution  ;  by  the  aid  of  which  we  shall,  a  little  later, 
proceed  to  calculate  the  resultant  of  any  number  of  forces  in  one 
plane.     We  might  calculate  the  resultant  by  applying  the  elementary 


15° 


ABSTRACT  DYNAMICS. 


formulae  (§§  419,  420,  42 1)  to  repetitions  of  the  parallelogram  of  forces. 
But  this  process  would  be  very  complicated  and  tedious,  if  the  forces 
were  numerous,  and  their  magnitudes  and  angles  given  in  numbers; 
and  we  shall  see  that  it  may  be  avoided  by  resolving  all  the  forces 
along  two  lines  at  right  angles  to  one  another,  and  thus  obtaining 
as  equivalent  to  them,  two  forces  along  these  lines. 

We  shall  first  consider  the  general  inverse  problem  (§  418),  or  the 
resolution  of  forces. 

431.  If  a  force  acting  on  a  material  point,  and  two  lines  in  one 
plane  with  the  line  of  that  force,  be  given,  it  is  possible  to  find  deter- 
minately  two  forces  along  those  lines,  of  which  the  given  force  is  the 
resultant. 

The  two  forces  thus  determined  are  called  the  components  of  the 
given  force  along  the  given  lines,  and  if  we  substitute  these  two  forces 
for  the  given  force,  we  are  said  to  resolve  the  given  force  into  two 
forces  along  the  given  lines;  or,  to  resolve  the  force  along  the  gfven 
lines. 

432.  Geometrical  Solution.  Let  M  be  the  given  point ;  R.,  the 
given  force  acting  on  it  in  the  line,  MK \ 
and  MFzx\<i  MG  the  given  lines. 

It  is  required  to  find  the  components 
along  MF2x\A  MG  of  i?  in  MK. 

Take  any  convenient  length  MD  to 
represent  the  magnitude  of  the  given 
foice,  R.  Through  D  draw  DA  parallel 
to  GM,  and  let  it  cut  MF  in  A  ;  and 
-also  through  D  draw  DB  parallel  to  FM,  and  let  it  cut  MG  in  B  \ 
MA  and  MB  represent  the  required  magnitudes  of  the  components. 

433.  Tngonometrical  Solution.  If  the  angle  KMF  be  given  =  a, 
and  KMG  =  [i,  and  if  the  required  component  of  the  given  force  R 
along  AIF  be  denoted  by  P,  and  the  component  along  MG  by  Q,  we 
deduce  from  equations  (2)  and  (3)  (§  420),  the  followmg : — 

sin(a  +  j8)'  ^'^^ 

R  sin  a 
sin  (a  +  p)' 

^34.     IVhen  the  given  lines  of  resolution  are  at  right  angles  to  one 
another,  these  expressions  are  modified  in  the  manner  shown  above 
(§  429,  Case  VII),  or  we  may  find  them 
at  once  from  the  geometrical    construc- 
j^/""  tion  proper  for  the  case,  thus: — 

Let   j\fX,   MY    be   the   given   lines; 
XMY  =  r,o\  and  MD  =  R.     Also,  as  be- 
fore, DMA  =  a,  and  DMB  =  p.     Draw 
.^      DA  parallel  to  YM,  or  perpendicular  to 
MX,  and  make  MB  =  AD.     Then  in  the 


Q- 


(18; 


STA  TICS   OF  A   PARTICLE.— A TTRA  CTIO.Y.       151 

right-angled  triangle  MAD,  MA^MDco-.DMA,  and  AD  =  MD 
sin  DMA. 

Hence,  since  MA  represents  the  component  along  MX,  and  MB 
the  component  along  MY, 

F^Rcosa,  (ig) 

^  =  i?  sin  a,  or  (2  =  ^  cos /?.  (20) 

Hence,  in  rectangular  resolution,  the  component,  along  any  line, 

of  a  given  force,  is  equal  to  the  product  of  the  number  expressing  the 

given  force,  into  the  cosine  of  the  angle  at  which  its  direction  is  in- 

clined  to  that  line. 

435.  Application  of  the  Resolution  of  Forces,  Let  there  be  a 
number  offerees  F^,  F^,  F^,  F^,  F^, 
acting  respectively  in  lines  ML^, 
ML  J,  ML^,  ML^,  ML^,  on  a  ma- 
terial point  M\  required  their  re- 
sultant. 

Through  ^^,  draw  at  right  angles 
to  each  other,  and  in  the  same  plane 
as  the  given  forces,  two  lines,  XX' 
and  YY' ,  which  maybe  called  lines 
or  axes  of  resolution.  Let  the  angle 
which  the  resultant  forms  with  the 
line  of  resolution  MX,  be  denoted 
by  B,  and  let  the  angles,  which  the 
lines  of  the  forces  make  respectively 

with  the  lines  of  resolution,  be  denoted  by  <x^,  P/,  a,,  /?, ;  a^ 
that  is,  L^MX^a^,  L^MY=(3^,  and  so  on. 

The  angles  [3^  ^^,  See,  are  merely  the  complements  of  a^  a^,  &c., 
and,  except  for  the  sake  of  symmetry,  they  need  not  have  been  intro- 
duced into  our  notation. 

Resolve  (§  434)  the  first  force  F^,  into  two  components,  one  along 
AIX,  and  one  along  MV.     These  are 

F^  cos  a,  along  MX,  which  force  may  be  denoted  by  .Y,. 
and     f,  sin  a^  along  MY,  which  force  may  be  denoted  by  F,. 

Treat  all  the  other  forces  in  like  manner,  thus  reducing  them  to 
components  along  MX  and  MV;  and  add  together  the  components 
along  each  of  the  lines  of  resolution.  Then  if  X  denote  the  sum  of 
the  components  along  MX,  and  Fthe  sum  of  the  components  along 
MV,  we  have 

X  =  Z',  cos  a,  +  F^  cos  a^  +  F^  cos  a^  +  F^  cos  a^  +  Fi,  cos  a,, 
y=  F,  sin  a,  +  F„  sin  a+F,  sin  a,  +  F,  sin  a  +  F.  sin  a.. 


f3^;Scc.; 


and 


Lastly,  to  find  the  resultant  of  X  and  K 
(§429).  R  =  >J(X'^V^, 

X 
R' 


cos  t^  =  • 


(2t) 
(22) 


,52  ^  abstract' by  NAifncs? 

or,  as  is^in  general  better  for  calculation; 


tan^--^,  (23) 

whence  we  derive  the  magnitude  of  the  resultant, 

R^Xsccd.  (24) 

The  calculation  will  in  general  be  facilitated  by  the  use  of  log- 
arithms; forwhich  purpose  equations  (23)  and  (24)  are  to  be  modified 
in  the  following  manner  : — 

tab.  log.  tan  9  ^  log.  Y-  log.  X+  10,  (25) 

log.  7?  =  log.  X'r  tab.  log.  sec.  6-  10.  (26) 

Remark  i. — It  is  to  be  observed  that  the  sums  X  of  the  different 
components  X,,  X.^,  &c.,  and  Fof  F,,  Y^,  Sec,  are  got  by  an  algebraic 
addition,  whatever  may  be  the  algebraic  signs  of  the  several  terms. 
If  the  given  forces  act  all  round  the  point  M,  it  will  happen  in  the 
resolution  that  the  different  components  do  not  all  act  in  the  same 
directions  along  XX'  and  YY'.  It  will  be  necessary,  therefore,  to 
fix  upon  one  direction  as  positive.  Thus,  if  AfX  and  A/Y  be  posi- 
tive directions,  AfX',  AfY',\vi\l  be  negative;  and  absolute  values  of 
the  components,  which  act  from  Af  to  X',  and  from  Af  to  Y\ 
must  be  subtracted  from,  instead  of  added  to,  those  along  AfX 
and  AfY 

Remark  2. — In  choosing  the  axes  of  resolution,  it  simplifies  the 
problem  to  fix  on  one  of  the  lines  which  represent  the  forces,  as  one 
of  the  axes,  and  a  line  perpendicular  to  it,  as  the  other. 

Let  AfL^,  the  line  of  the  first  force  /',,  be  the  axis  A/X,  and  AfY, 
a  line  perpendicular  to  it,  the  other, 

a,  in  this  case  is  nothing;  and  the  angle  P^  AfP^  -a^. 
.    Hence,  if  a,  =  o,  the  resolution  of  the  first  force  is 

^  1^1  =  ^1  cos  ai^/'i, 

'  \  K,  =  /\  sin  Oj  =  o, 

that  is,  /',  requires- no  resolution. 

If  two  of  the  forces  happen  to  be  at  right  angles,  it  will  be  con- 
venient to  choose  the  axes  along  them,  and  then  neither  requires 
resolution. 

Actual  cases  may  often  be  simplified  by  observing  if  any  two  of  the 
forces  are  opposite,  in  which  case,  one  force,  equal  to  the  excess  of 
the  greater  above  the  less,  and  acting  in  the- direction  of  the  greater, 
may  be  taken  instead  of  them. 

Remark  3. — When  the  direction  of  the  resultant  is  known,  and  its 
magnitude  is  required,  it  is  most  convenient  to  make  it  one  of  the 
axes  of  resolution. 


STATICS  OF  A   PARTICLE.— ATTRACT/ON.      153 

'■  Let  MK  be  the  direction  of  the 
resultant  of/-,,  /', ,  /",,  P^,  the  dif- 
ferent forces.  Resolve  each  force 
into  two,  one  along  MK,  and  one 
in  a  line  perpendicular  to  it.  Add 
the  components  along  MK.  The 
sum  must  be  the  magnitude  of  the 
resultant;  and  the  components  along 
the  other  line  must  balance  one  an- 
other.    Hence, 

X=R=P^cos  A,  MK  +  P^  cos  A^  MK  +  &c, 
and  F=  P^  sin  A^  MK  +  Z',  sin  A,  MK  +  &c.  =  0. 

Remark  4. — Equations  (23)  and  {24)  may'be  employed  with  ad- 
vantage in  all  cases  where  the  numbers  of  significant  figures  in  the 
values  to  be  used  for  X  and  Fare  large. 

By  equations  (23)  and  (24)  the  direction  of  the  resultant  is  first 
determined,  and  then  its  magnitude,  not  as  in  equations  (21)  and  (22), 
the  magnitude  first,  and  then  the  direction. 

436.  For  the  better  understanding  of  what  follows  a  slight  digres- 
sion (§§  437,  464)  upon  projections  and  geometrical  co-ordinates  is 
now  inserted. 

437.  The  projection  of  a  point  on  a  straight  line,  is  the  point  in 
which  the  latter  is  cut  by  a  perpendicular  to  it  from  the  former. 

438.  Any  line,  joining  two  points,  is  called  an  arc.  It  is  not 
necessary  to  confine  this  expression  to  its  most  usual  signification  of 
a  continuous  curve  line.  It  may  be  applied  to  a  straight  line  joining 
two  points,  as  an  extreme  case;  or  it  may  be  applied  to  a  zigzag  or 
angular  path  from  one  point  to  the  other;  or  to  a  self-cutting  path, 
whether  curved  or  polygonal;  in  short,  to  any  track  whatever,  from 
one  point  to  the  other. 

439.  The  projection  of  an  arc  on  a  straight  line,  is  the  portion  of 
the  latter  intercepted  between  the  projections  of  the  extremities  of  the 
former. 

440.  If  we  imagine  an  arc  divided  into  any  number  of  parts,  the 
projections  of  these  parts,  taken  consecutively  on  any  straight  line, 
make  up  consecutively  the  projection  of  the  whole.  Hence,  the  sum 
of  the  projections  of  the  parts  is  equal  to  the  projection  of  the  whole. 
But  in  this  statement,  it  must  be  understood  that,  of  such  partial 
projecdons  laid  down  in  order,  those  which  are  drawn  in  one  di- 
rection, or  forwards,  being  reckoned  as  positive,  those  which  are 
drawn  in  the  other  direcuon,  or  Iku:/: wards,  must  be  reckoned  as 
negative. 

441.  The  prpjection  of  an  arc  on  any  straight  line,  is  equal  to  the 
length  of  the  straight  line  joining  the  extremities  of  the  former,  mul- 


154  ABSTRACT  DYNAMICS. 

tiplied  by  the  cosine  of  the  angle'  at  which  it  is  inclined  to  the  latter. 
This  angle,  if  not  a  right  angle,  v/ill  be  acute  or  obtuse,  according  fo 
the  convention  which  is  understood  as  to  the  direction  reckoned 
positive  \n  the  line  of  projection ;  and  the  extremity  of  the  arc  which 
is  izktn  first  in  drawing  z.  positive  line  from  one  extremity  of  it  to  the 
other. 

442.  The  orthogonal  projection  of  a  line,  straight  or  curved, 
closed  or  not  closed,  on  a  plane,  is  the  locus  of  the  points  in  which 
the  latter  is  cut  by  perpendiculars  to  it  from  all  points  of  the  former. 

Other  kinds  of  projections  are  also  used  in  geometry;  but  when  no 
other  designation  is  applied  or  understood,  the  simple  term  projection 
will  always  mean  orthogonal  projection. 

443.*^  A  circuit  is  a  line  returning  into  itself,  or  a  line  without  ends 
in  a  finite  ^ace.  It  is  (if  a  continuous  curve)  often  called  a  dosed 
curve;  or  if  made  up  altogether  of  rectilinear  parts,  a  closed  polygon. 
A  circuit  in  one  plane  may  be  either  simple  or  self-cutting.  The  latter 
variety  has  been  called  by  De  Morgan,  autoiomic.  But  whether  simple 
or  autotomic,  there  is  just  one  definite  course  to  go  round  a  circuit; 
and 'at  double  or  multiple  points,  this  course  must  be  distinctly 
indicated'  (arrow-heads  being  generally  used  for  the  purpose  on 
a' diagram,  like  the  finger-posts  where  two  or  more  roads  cross).  A 
circuit  not  confined  to  one  plane  need  never  be  considered  to  be 
autotomic,  unless  as  an  extreme  case.  Thus,  if  we  take  any  thread  or 
wire,  however  fine,  and  bend  it  into  any  curve  or  broken  line,  or  tie  it 
into  the  most  complicated  knot  or  succession  of  knots,  but  attach  its 
ends  together;  any  geometrical  line  drawn  altogether  within  it,  from 
any  one  point  of  it.  round  through  its  length  back  to  the  same  point, 
constitutes  essentially  a  simple  or  not  self-cutting  circuit. 

444.  'The  area  enclosed  by,'  or  'the  area  of  a  simple  plane 
circuit,  is  an  expression  which  requires  no  explanation.  But,  as  has 
been  shown  by  De  Morgan^,  a  peculiar  rule  of  interpretation  is 
necessary  to  apply  the  same  expression  to  an  autotomic  plane  circuit, 
and  it  has  no  application,  hitherto  defined,  to  a  circuit  not  confined  to 
one  plane. 

445.  The  area  of  an  autotomic  plane  circuit,  is  the  sum  of  the 
areas  of  allots  parts  each  multiplied  by  zero  with  unity  as  many  times 
added  as  the  circuit  is  crossed''  from  right  to  left,  and  unity  as  many 

1  The  angle  at  which  one  line  is  inchncd  to  another,  is  the  angle  between  two 
lines  drawn  parallel  to  them  from  any  point,  in  directions  similar  to  the  directions 
m  the  given  lines  which  are  reckoned  positive. 

'  'A  curve  which  has  double  or  multiple  points,  may  be  in  many  different 
ways  a  circuit,  or  mode  of  proceeding  from  one  point  to  the  same  again.  Thus  the 
-figure  of  8  may  be  traced  as  a  ulj-cutting  circuit,  in  the  way  in  which  it  is  natural 
if  the  curve  be  a  continuous  lennnscatr,  or  it  may  be  traced  as  a  circuit  presenting 
two  coincident  salient  points.  A  determinate  area  requires  a  determmateniode  of 
making  the  circuit.'  De  Morgan,  Camhrvigc  and  Dublin  Alathematual  yournal, 
May,  1S50. 

^  '  Exten:,ion  of  the  word  area,'  Cambridge  and  Dublin  Mathematical  yoitynal. 
May,  1850. 

*  A  rr.oving  point  is  said  to  cross  a  plane  circuit  from  right  to  left,  if  it  crosses 


STATICS   Or  A  PARTICLE.— ATTJ^ action:     155 

'times  subtracted  as  the  circuit  is  crossed  from  left  to  right,  when  a 
point  is  carried  in  the  plane  from  the  outside  to  any  position  within 
the  enclosed  area  in  question.  The  diagram,  which  is  that  given  by  De 
Morgan,  will  show  more  clearly  what  is  meant  by  this  use  of  the  word 
area.  The  reader,  with  this  as  a  model,  may  exercise  himself  by 
drawing  autotomic  circuits  and  numbering  the  different  portions  of 
the  enclosed  area  according  to  the  rule,  which  he  will  then  find  no 
difficulty  in  understanding. 


446.  Any  portion  of  surface,  edged  or  bounded  by  a  circuit,  is 
called  a  shelL 

A  plane  area  may  be  regarded  as  an  extreme  case,  but  generally 
the  surface  of  a  shell  will  be  supposed  to  be  curved. 

A  simple  shell  is  a  shell  of  which  the  surface  is  single  throughout. 
One  side  of  the  shell  must  always  be  distinguished  from  the  other, 
whatever  may  be  the  convolutions  of  its  surface.  Thus  we  shall  have 
a  marked  and  unmarked  side,  or  an  outside  and  an  inside,  to  dis- 
tinguish from  one  another. 

447.  The  projection  of  a  shell  on  any  plane,  is  the  area  included 
in  the  projection  of  its  bounding  line. 

448.  If  we  imagine  a  shell  divided  into  any  number  of  parts,  the 
projections  of  these  parts  on  any  plane  make  up  the  projection  of  the 
whole.  But  in  this  statement  it  must  be  understood  that  the  areas  of 
partial  projections  are  to  be  reckoned  as  positive  only  if  the  mark(.d 
side,  or,  as  we  shall  call  it,  the  outside,  of  -the  projected  area,  and  a 
marked  side,  which  we  shall  call  the  front,  of  the  plane  of  projection, 
face  the  same  way. 

If  the  outside  of  any  portion  of  the  projected  area  faces  on  the 
whole  backwards,  relatively  to  the  front  of  the  plane  of  projection,  the 
projection  of  this  portion  is  to  be  reckoned  as  negative  in  the  sum. 

from  the  right  side  to  the  left  side  as  regarded  by  a  person  lcx)king  from  any  poini 
oithe  circuit  in  the  direction  reckoned  positive. 


156  ABSTRACT  DYNAMICS. 

Of  course  if  the  projected  surface,  or  any  part  of  it,  be  a  plane' area 
at  right  angles  to  the  plane  of  projection,  the  projection  van^hes. 

Cor.  The  projections  of  any  two  shells  having  a  common  edge,  on 
any  plane,  are  equal.  The  projection  of  a  closed  surface  (or  a  shell 
with  evanescent  edge),  on  any  plane,  is  nothing. 

449.  Equal  areas  in  one  plane  or  in  parallel  planes,  have  equal 
projections  on  any  plane,  whatever  may  be  their  figures.  [The  proof 
is  easily  found.] 

Hence  the  projection  of  any  plane  figure,  or  of  any  shell,  edged  by 
a  plane  figure,  on  another,  plane,  is  equal  to  its  area,  multiplied  by  the 
cosine  of  the  angle  at  which  its  plane  is  inclined  to  the  plane  of  pro- 
jection. This  angle  is  acute  or  obtuse,  according  as  the  marked  sides 
of  the  projected  area,  and  of  the  plane  of  projection  face,  on  the  whole, 
towards  the  same  parts,  or  on  the  whole  oppositely. 

450.  Two  rectangles,  with  a  common  edge,  but  not  in  one  plane, 
have  their  projection  on  any  other  plane,  equal  to  that  of  one  rect- 
angle, having  their  two  remote  sides  for  one  pair  of  its  opposite  sides. 
For,  the  sides  of  this  last-mentioned  rectangle  constitute  the  edge  of 
a  sAe//,  which  we  may  make  by  applying  two  equal  and  parallel 
triangular  areas  to  the  sides  of  the  given  rectangles;  and  the  sum  of 
the  projections  of  these  two- triangles  on  any  plane,  according  to  the 
rule  of  §  448,  is  nothing. 

Hence  (as  is  shown  by  a  very  simple  geometrical  proof,  which  is 
left  as  an  exercise  to  the  student),  we  have  the  following  construction 
to  find  a  single  plane  area  whose  projection  on  any  plane  is  equal  to 
the  sum  of  the  projections  of  any  two  given  plane  areas. 

From  any  convenient  point  of  reference  draw  straight  lines  per- 
pendicular to  the  two  given  plane  areas  forward,  relatively  to  their 
marked  sides  considered  as  fronts.  Make  these  Imes  numerically 
equal  to  the  two  areas  respectively.  On  these  describe  a  parallel- 
ogram, and  draw  the  diagonal  of  this  parallelogram  through  the  point 
of  reference.  Place  an  area  with  one  side  marked  as  front,  in  any 
position  perpendicular  to  this  diagonal,  facing  forwards,  and  relatively 
to  the  direction  in  which  it  is  drawn  from  the  point  of  reference. 
Make  this  area  equal  numerically  to  the  diagonal.  Its  projection  on 
any  plane  will  be  equal  to  the  sum  of  the  projections  of  the  two  given 
areas,  on  the  same  plane. 

The  same  construction  maybe  continued;  just  as,  in  §  413,  the 
geometrical  construction  to  find  the  resultant  of  any  number  of 
forces;  and  thus  we  find  a  single  plane  area  whose  projection  on  any 
plane  is  equal  to  the  sum  of  the  projections  on  the  same  plane  of  any 
given  plane  areas.  And  as  any  shell  may  (if  it  be  not  composed  of  a 
finite)  be  regarded  as  composed  of  an  infinite  number  of  plane  areas, 
the  same  construction  is  applicable  to  a  shell.  Hence  the  projection 
of  a  shell  on  any  plane  is  equal  to  the  projection  on  the  same  plane, 
of  a  certain  plane  area,  determined  by  the  preceding  construction. 

From  this  it  appears  that  the  projection  of  a  shell  is  nothing  on 


STATICS   OF  A   PARTICLE.— ATTRACTION.       157 

any  plane  perpendicular  to  the  one  plane  on  which  its.  projection 
is  greater  than  on  any  other;  and  that  the  projection  on  any  inter- 
mediate plane  is  equal  to  the  greatest  projection  multiplied  by  the 
cosine  of  the  inclination  of  the  plane  of  the  supposed  projection  to 
the  plane  of  greatest  projection. 

451.  To  specify  a  point  is  to  state  precisely  its  position.  As  we? 
have  no  conception  of  position,  except  in  so  far  as  it  is  relative,  the 
specification  of  a  point  requires  definite  objects  of  reference,  that  is, 
objects  to  which  it  may  be  referred.  The  means  employed  for  this 
purpose  are  certain  elements  called  co-ordinates,  from  the  system  of 
specification  which  Descartes  first  introduced  into  mathematics.  This 
system  seems  to  have  originated  in  the  following  method,  for  de- 
scribing a  curve  by  a  table  of  numbers,  or  by  an  equation. 

452.  Given  a  plane  curve,  a  fixed  line  in  its  plane,  and  a  fixed 
point  in  this  line,  choose  as  many  points  in  the  cur\-e  as  are  required 
to  indicate  sufficiently  its  form ;  draw  perpendiculars  from  them  to 
the  fixed  line,  and  measure  the  distances  along  it,  cut  off  by  these 
lines,  reckoning  from  the  fixed  point.  In  this  way  any  number  of 
points  in  the  curve  were  specified.  The  parts  thus  cut  off  along  the 
fixed  line,  were  termed  iincac-  al'sassac,  and  the  perpendiculars,  lincae 
ordinatim  applicatae.  The  S)-stem  was  afterwards  improved  by  draw- 
ing through  the  point  of  reference  a  line  at  right  angles  to  the  first, 
and  measuring  off  along  it  the  ordmaies  of  the  curve.  The  two  lines 
at  right  angles  to  one  another  are  called  the  axes  of  reference,  or  the 
lines  of  reference.  The  ordinate  and  abscissa  of  any  point  are  termed 
its  co-ordinates;  and  an  equation  between  them,  by  which  either  may 
be  calculated  when  the  other  is  eiven,  expresses  the  curve  in  a  per- 
fectly full  and  precise  manner. 

453.  It  is  not  necessary  that  the  lines  of  reference  be  chosen  at 
right  angles  to  each  other.  But  when  they  are  chosen,  inclined  at 
any  other  angle  than  a  right  angle,  the  co-ordinates  of  the  point 
specified  are  not  its  perpsodicular  distances  ^rom  them,  but  its 
distances  from  either,  measured  parallel  to  the  other.  Such  oblique 
co-ordinates  .are  sometimes  convenient,  but  rectangnlar  co-ordinates 
are,  in  general,  the  most  useful;  these  we  shall  now  consider. 

454.  If  the  points  to  be  specified  are  all  in  one  plane,  the  objects 
of  reference  are  two  lines  at  right  angles  to 

one  another  in  that  plane.  Thus,  let  P 
be  a  point  in  a  plane  XOY;  and  let  OX, 
OY,  be  two  lines  in  the  plane,  cutting  each 
other  at  right  angles  in  the  point  O.  Then  B 
will  the  position  of  the  point  P  be  known, 
if  the  perpendicular  distance  of  the  point 
P  from  the  line  OX,  namely,  the  length 
of  the  line  PA,  and  the  perpendicular  dis-  Q^ 
tance  from  0  V,  namely,  the  length  of  the 
line  PP,  be  known. 


i58 


ABSTRACT  DYNAMICS'. 


455.  Again,  let  points,"  not  all  in  one 
plane,  but  in  any  positions  through  space 
be  considered.  To  specify  each  point  now, 
three  co-ordinates  are  required,  and  the 
objects  of  reference  chosen  may  be  three 
planes  at  right  angles  to  one  another;  thus, 
the  point  P  is  specified  by  the  lines  PK, 
PJI,  PI,  drawn  perpendicular  to  the  planes 
YZ,  ZX,  XY,  respectively. 

In  our  standard  diagrams  'the  positive 
directions  OX,  OY,  OZ,  are  so  taken  that 
if  a.  watch  is  held  in  the  plane' A^ (9  F,  with  its  face  towards  OZ,  an 
angular  motion  againsf  ( he  hands  would  carry  a  line  from  OX  \q  O  Y, 
through  the  right  angle  XOY. 

.456.  When  the  objects  to  be  specified  are  lines  all  passing  through 
one  point,  the  specifying  elements  employed,  are  angles  standing  in 
definite  relation  to  them,  and  to  the  objects  of  reference.  There  are 
two  chief  modes  in  which  this  kind  of  specification  is  carried  out: 
"he  polar  and  the  symmetrical, 

457.  Polar  Method.  i°.  Lines  all  in  one  plane.  In  this  case  the 
object  of  reference  is  any  fixed  line  through  their  common  point  of 
intersection,  and  lying  in  their  plane. 

Let  O  be  the  common  point  of  intersection, 
^ — "^       OX  the  fixed   line,  and    OP  the  line   to   be 

Q  ~- — '         X      specified.     Then  the  position  of  OP  will  be 

known,  if  the  angle  XOP,  which  the  line  OP 
makes  with  OX,  be  known. 

2°.  Lines  in  space,  all  passing  through  one  point,  may  be  specified 
by  reference  to  a  plane  and  a  line  in  it,  both  passing  through  their 
common  point  of  intersection. 

Let  OP  be  one  of  a  number  of  lines,  all 
passing  through  O,  to  be  specified  with  refer- 
enced to  the  plane  XOY,  and  the  line  OX  in  it. 
Through  OP  let  a  plane  be  drawn,  cutting  the 
plane  X(9 Fat  right  angles  in  OE.  Then  the 
line  OP  will  be  specified,  if  the  angles  XOE, 
£0P  arc  given. 

Corollary.  Similarly,  if  the  line  OP  be  the 
locus  of  a  scries  of  points,  any  one  of  these 
points  will  be  specified,  if  its  distance  from  O 
and  the. two  angles  specifying  the  line  OP,  are 
known. 

458.  Synvnelrical  Method.  In  this  method  the  objects  of  reference 
are  three  lines  at  right  angles  to  each  other,  through  the  common 
point  of  intersection  of  the  lines  to  be  specified,  and  the  specifying 
elements' are  the  three  angles  which  each  line_  makes  with  these  three 
lines  of  reference. 

Thus,  if  O  be  the  common  point  of  intersection,  OK  one  of  the 


STATICS   OF  A    PA RT/CLE— ATTRACTION.      159 

lines  to  be  specified,  and  OX,  OV,  OZ,  the  linos  of  reference;  then 
the  angles  XOK,   YOK,  ZOK,  are  the  specifying  elements. 

459.  From  what  has  now  been  said,  it  will  be  seen  that  the  pro- 
jections of  a  given  line  on  other  three  at  right  angles  to  each  other 
are  immediately  expressible,  if  its  direction  is  specified  by  either  of 
the  two  methods. 

1°.  Polar  Method.  Let  OK  be  the  given  line, 
and  OX,  OV,  OZ,  the  lines  along  which  it  is  to  be 
projected.  Through  OZ  and  OA'  lot  a  jjlane  pass, 
cutting  the  plane  XOV  in  OP.  Through  A'  draw 
another  plane,  KEA,  cutting  OX  perpendicularly  in 
A  and  KEB  cutting  OY  perpendicularly  in  B. 
Then  KE,  being  the  intersection  of  two  planes 
each  perpendicular  to  XOY,  is' perpendicular  to 
every  line  in  this  plane.  Hence,  OEK  is  a  right 
angle. 

Hence, 

OE  -  OK  cos  KOE. 

Again,  since  the  plane  KAE  was  drawn  perpendicular  to  OX, 
OAE  is  a  right  angle. 

Hence,  OA  =  OE  cos  EOX  =  OK  cos  KOE  cos  EOX, 

and  similarly,    OB  =  OE  cos  EO  K=  6* A' cos  KOE  cos  EO  Y, 
or  if  we  put 

OK=r,    XOE=<i>,   EOK=!,   KOZ^Q^\iT-i, 

and  let  the  required  lines  be  denoted  by  .v,  y,  z,  then 
x  =  r  sin  0  cos  t/),j 

y  -  )■  sin  0  sin  (^.>  (i) 

z  =  r  cos  9.  ) 

2°.    Sy>j:»icfn'cal  Method.     Let  the  line  he  referred  to  rectangular 
axes  by  the  three  angles,  a  ^  KOX,   ft  -  KO  Y,    y  =  KOZ. 
Then  the  required  projections  are 

X  -  r  cos  a,    y  -r  cos  /?,    z  -  r  cos  y. 

460.  Referring  again  to  the  diagram,  we  have 

0E'=  OA'+OB\ 
and  OK'=OE'+OC\ 

therefore,  OK'-  =  OA' +  OB'  +  0C\ 

or  r^  =  x'  +y^  +  z\  (:) 

Substituting  here  for  .r,  y,  z,  their  values,  in  terms  of  r,  a,  (3,  y, 
found  above,  and  dividing  both  members  of  the  resulting  equation  by 
r',  we  have 

I  =  cos' a  +  cos' /J  4- COS"  y.  (j) 

461.  In  the  symmetrical  method,  three  angles  arc  used;  but,  as  we 
have  seen,  only  two  are  necessary  to  fix  the  position  of  the  line.     \\'e 


Lie  ABSTRACT  DYNAMICS. 

now  see'that,  if  two  of  the  three  angles,  a,  /?,  y,  are  gfven,  the  third  ^ 
can  be  found.     Suppose  a  and  ^  given,  tiien  by  §  460,- 
cos*  y  =  I  -  cos^  a  -  cos^  y3. 

For  logarithmic  calculation,  the  following  modification  of  the  pre- 
ceding formula  is  useful, 

COS^  y  -  sin^  a  -  cos'  /3  =  -r  COS  (a  +  /3)  x  COS  (a  -  /3), 
whence        cos  y  =  ^\-  cos  (a  +  yS)  x  cos  (a  -  /?)} 

=  \/{cos  (tt  -  a  -  jS)  X  cos  (a  -  jS)}, 
Tab.  Log.  cos  y  =  ^-  {T.  L.  cos  (tt  -  a  -  /3)  +  T.  L.  cos  (a  -  /3)}.        (4) 

462.  The  following  comparison  will  show  in  what  way  the  two 
systems  are  related,  and  how  it  is  possible  to  derive  the  specifying 
elements  of  either  from  those  of  the, other.  In  the  polar  method,-  the 
fixed  line  in  the  equatorial  plane,  corresponds  with  one  of  the  three 
lines  of  reference  in  the  symmetrical.  A  line  in  the  equatorial  plane, 
drawn  at  right  angles  to  the  fixed  line  of  the  polar  system,  constitutes 
a  second  line  of  reference  in  the  symmetrical  system.  The  third  line 
in  the  symmetrical  system,  is  the  axis  of  the  polar  system,  from 
which  the  polar  distance  {&)  is  measured.  A  comparison  of  pre-' 
ceding  formulae  shows  that 

cos  a  =sin  5  cos  <^\ 

cos /?  =  sin  ^  sin  <^,  >  (5) 

cos  y  =  cos  Q,  ) 

463.  The  cosines  of  the  three  angles,  a,  /3,  y,  of  the  symmetrical 
system,  are  commonly  called  the  direction  cosines  of  the  line  specified. 
If  we  denote  them  by  /,  in,  n,  we  have  as  above, 

/-  +  OT°+  ?i^  =  I,  (6) 

A  line  thus  specified  is  for  brevity  called  the  line  (/,  ni,  n). 

If/,  m,  n,  are  the  direction  cosines  of  a  certain  line;  it  is  clear  that 
-/,  -  m,  - //,  are  the  direction  cosines  of  the  line  in  the  opposite 
direction  from  O.  Thus  it  appears  that  the  direction  cosines  of  the 
line,  specify  not  only  the  straight  line  in  which  it  lies,  but  the  direction 
in  it  which  is  reckoned  as  positive. 

464.  We  conclude  this  digression  with  some  applications  of  the 
principles  explained  in  it,  which  are  useful  in  many  dynamical  in- 
yestig^g-tions. 

{a)  To  find  the  mutual  inclination,  6^  of  two  lines,  (/,  7n,  n), 
{f,  tn,  n).  Measure  off  any  length  OK=r,  along  the  first  line  (see 
fig.  of  §  459).     We  have,  as  above, 

OA^^lr,    AE  =  fnr,    EK=nr. 

Now  (§  441),  the  projection  of  OK  on  the  second  line,  is  equal  to 

the  sum  of  the  projections  of  OA,  AE,  EK,  on  the  same.     But  the 

cosines  of  the  angles  at  which  these  several  lengths  are  inclined  to  the 

line  of  projection,  are  respectively  cos  9,  H ,  rd,  n.     Hence 

CA'cos  Q  =  OAr  +  A  Em'  +  EKn'. 


STATICS   OF  A   PARTICLE.— ATTRACTION.      \t^ 

If  we  substitute  in  this,  for  OK,  OA,  AE,  EK,  their  values  shown 
above,  arid  divide 'both  members  by  r,  it  becomes 

cos  B  -  U'  +  nun  +  7in\  (7) 

a  most  important  and  useful  formula. 

Sometimes  it  is  useful  to  have  the  sine  instead  of  the  cosine  of  0. 
To  find  it  we  have  of  course, 

sin'  ^  =  I  -  (//'.+  7nm'  ^nn'y. 

This  expression  may  be  modified  thus: — instead  of  i,  take  what  is 
equal  to  it,  {r  +  m'^  -f  ;/-)  (/-  +  7/r  +  rr), 

and  the  second  member  of  the  preceding  becomes 

(/•  +  ;/r  +  ir)  (/^  +  ;//''  +  ;/■')  -  (//'  +  ;//;;/'  -f  nn')- 

=  [nin'Y  -r  {'ini')'  —  2inni')i;i  +  &:c.  =  {nui  —  7V?i'Y  -f  o:c, 

IlencCj         sin  6  -  \{inn'  -  nm')'  +  (///"  -  ///)"  4-  {lui  —  vil' f\-.  (S) 

(/•)  To  find  the  direction  cosines,  A,  /.i,  r,  of  the  common  perpen- 
dicular to  two  lines,  (/,  ;;;,  ;/),  (/',  'i'.',n'). 

Tiie  cosine  of  the  inclination  of  (A,  jx,  \)  to  (/,  ;//,  n)  i.s,  according 
to  (7)  above,  /\  +  piij.  +  /iv;  and  therefore 

/\  +  ;/;  /x  +  7/  V  =  0.. 
similarly  /'A  +  ju'ij.  +  n'v  ~  o.'^  (9) 

also  (.§  463)  A"  +  fi"-  +  v-  =  I.) 

These  three  equations  suffice  to  determme  the  three  unknoAvu 
quantities,  A,  /x,  v.     Thus,  from  tlie  first  two  of  them,  we  have 

A  w.  V 

7/1  n  —  nm       711  —  in       l/n  - 1  in  ' 

From  these  and  the  third  of  (9),  we  concluGc 

\  = _ -^ ^ ^ _ _^-,    ^-c., 

\{uvi  —7'jn'f  +  [ill' -  in')-  +  (^Iin' - 7Ki )'\- 

or  if  we  denote,  as  above,  by  6,  the  mutual  inclination  of  (/,  w,  n) 
{f,  in,  It); 

(mil'  —  itvi')              (n/' -//.'')  Uiti  -  nil')                ,     , 

sm  ^            ^         SHI  ^     '  sm  0                         ' 

The  sign  of  each  of  these  three  expressions  may  be  changed,  in  as 
much  as  either  sign  may  be  given  to  the  nu:nerical  value  found  for 
sin^  by  (8),  But  as  they  stand,  if  sin  0  is  taken  positive,  they  express 
the  direction  cosines  of  the  perpendicular  drawn  from"  O  through  the 
face  of  a  watch,  held  in  the  plane  (/,  ?>:,  ;;),  ('/',  ;//,  ;/').  and  so  facing 
that  angular  motion,  a,(;a!nst  or  icitli  ihc  iuir.ds,  would  carry  a  line 
from  the  direction,  (/,  m,  ;/),  through  an  angle  less  than  iSo'  to  the 
direction,  (/',  m',  71).  according  as  'angular  motion,  through  a  r;ght 
angle  from  OX  to  OY'v-^  against  or  with  tlic  hands  of  a  watch,  held  in 

\'ul.   -3— S 


1 62  ABSTRACT  DYNAMICS. 

the  plane  XOY,  and  facing  towards  OZ.  This 'rule  is  proved  by 
supposing,  as  a  particular  case,  the  lines  (/,  ;;/,  //),  (/',  in\  n'),  to 
coincide  with  OX  zxid.  6>  F  respectively;  and  then  supposing  them 
altered,  in  their  mutual  inclination  to  any  other  angle  between- o  and 
TT,  and  their  plane  turned  to  any  position  whatever. 

If  we  measure  off  any  lengths,  OK=r,  and  OK'  =  >  ,  along  the 
two  lines,  (/,  w,  Ji)  and  (/',  ;;/,  «'),  and  describe  a  parallelogram 
upon  them,  its  area  is  equal  to  rr  sin  6,  since  /  sin  6  is  the  length  of 
the  perpendicular  from  K'  to  OK.  Hence,  using  the  preceding 
expression  (8)  for  sin  6,  and  taking 

lr  =  x,        mr=y,        nr  =  z, 
Ir' —  x\     m'r'--y',     iir' —  z\ 

•we  conclude  the  following  propositions. 

{c)  The  area  of  a  parallelogram  described  upon  lines  from  tlie 
origin  of  co-ordinates  to  points  {x,y,  z),  {x',j,z')  is  equal  to 

{{yz'  -y'zY  +  (zx'  -  z'xf  +  {xy'  -  ;t»=}i.  (12) 

And,  as  X,  /x,  v,  are  the  cosines  of  the  angles  at  which  the  plane  of 
this  area  is  inclined  to  the  planes  of  YZ,  ZX,  XY,  respectively,  its 
projections  on  these  planes  are 

yz' —fz,     zx' —  z^x,     xy'  —  x'y.  (13) 

The  figures  of  these  projections  are  parallelograms  in  the  three 
planes  of  reference;  that  in  the  plane  YZ,  for  instance,  being  de- 
scribed on  lines  drawn  from  the  origin  to  the  points  (7,  z)  and  {y  ,  z). 
It  is  easy  to  prove  this  (and,  of  course,  the  corresponding  expressions 
for  the  two  other  planes  of  reference,)  by  elementary  geometry. 
Thus,  it  is  easy  to  obtain  a  simple  geometrical  demonstration  of  the 
equations  (8)  and  (11).  It  is  sufficient  here  to  suggest  this  investi- 
gation as  an  exercise  to  the  student.  It  essentially  and  obviously 
includes  the  ride  of  signs,  stated  above  (§  464  {a)). 

(d)  The  volume  of  a  parallelepiped  described  on  OK,  OK,  OK', 
three  lines  drawn  from  0  to  three  points 

{X,y,z),     {x',y\z'),     {x",y",z"),     is  equal  to 
x"  {yz'  —y'z)  +y"  {zx'  —  zx)  -i-  z"  {xy' -  x'y),  (14) 

an  expression  which  is  essentially  positive,  if  OK,  OK,  OK" ,  are 
arranged  in  order  similarly  to  OX,  OY,  (9Z(see  §  455  above).  The 
proof  is  left  as  an  exercise  for  the  student. 

In  modern  algebra,  this  expression  is  called  a  dderminant,  and  is 
■written  thus : — 

x',   y,    s',  (15) 

X  ,    y  ,     z  . 

46,5.  To  find  the  resultant  of  three  forces  acting  on  a  material 
point  in  lines  at  right  angles  to  one  another. 


M 'A 

in    the   plane   BAf/1 ; 


STATICS   OF  A  PARTTCLE.—ATTRACTIOy.      163 

1°.  To  find  the  magnitude  of  the  resultant. 
Let  the  forces  be  given  numerically,  X,  K,  Z, 
and  let  them  be  represented  respectively  by 
the  lines  AfA,  MB,  MC  at  right  angles  to 
one  another. 

First  determine  the  resultant  of  X  and  Y 
in  magnitude.  If  we  denote  it  by  R,  we 
have  (3429) 

i?'=,/(Z-Vr).  (i) 

This  resultant,  represented  by  ME,  lies 
and  since  tiic  lines  of  the  forces  X  and  Y  are  perpendicular  to  the 
line  MC,  the  line  ME  m.ust  also  be  perpendicular  to  it;  for,  if  a  line 
be  perpendicular  to  two  oilier  lines,  it  is  perpcndiadar  to  ei'ery  other  lin^ 
i?i  their  plane;  hence  R'  acts  perpendicularly  to  Z. 

Next,  find  the  resultant  of  K  and  Z,  the  third  force.  If  we  denote 
it  by  A^,  we  have 

and  substituting  for  R'  its  value,  we  have 

i?  =  ^/(X'  +  y^  +  Z^).  (2) 

2^  To  fi)id  the  direction  of  the  resultant.  Detefmine  first  the 
inclination  of  the  subsidiary  resultant  R  to  MA  or  MB  Let  the 
angle  EMA  be  denoted  by  0 ;  then  we  have 

cos  i'^-j^,^ 

Next,  let  y  denote  the  angle  at  which  the  line  MD  is  inclined  to 
MC)  that  is,  the  angle  CMD ;  we  have 

Z 


cos  y  -.  - 


A' 


(3) 


Thus,  by  means  of  the  two  angles  y  and  <;S,  the  position  of  the  line 
MD,  and,  consequently,  that  of  the  resultant  is  found. 

46G.  In  the  numerical  solution  of  actual  cases,  it  will  generally  be 
found  most  convenient  to  calculate  thetlirec  elements  in  the  following 
order:   1°,  c^,  2°,  y,  3-^,  R. 

1°.     To  calculate  cf>,  the  formula  already  given,  may  be  taken 

tan<^  =  ^.  (4) 


2'^.    To  calculate  y.     We  have 

R' 
tany  =  -^-. 


But 

Hence 

3°.    To  calculate  R. 


R  =  Xseci>. 

Xsecd* 
tan  y  =  — y~  , 

R  -  Zscc  y. 


(5) 
(7) 


1 64  ABSTRACT  DYNAMICS. 

467,  The  angles  determined  by  these  equations  speclfylhe'  line  of 
the  resultant,  by  what  was  called  in  previous  sections  (§§  457,  459) 
the  Folar  Method. 

The  symmetrical  specification  of  the  resultant  is  to  be  found  thus : 
Let  (in  fig.  of  §  465)  the  angles  at  which  the  line  of  the  resultant, 
MD,  is  inclined  to  those  of  the  forces  be  respectively  denoted  by  a,  /?, 
and  y.     Then,  as  above  (equation  (3)), 

Z 

cosy=-^.  (8) 

By  the  same  method  we  shall  find 

cosa--,  (9) 

Y 

and  cos/?  =  -^.  ro) 

If,  therefore,  there  are  three  forces  at  right  angles  to  one  another, 
the  cosine  of  the  inclination  of  their  resultant  to  any  one  of  them  is 
equal  to  this  force  divided  by  the  resultant 

This  method  requires  that  the  magnitude  of  the  resultant  be  known 
before  its  position  is  determined.  For  the  latter  purpose,  any  two  of 
the  angles,  as  was  shown  in  Chapter  V,  are  sufficient. 

468.  We  shall  now  consider  the  resolution  of  forces  along  three 
specified  lines.  The  most  important  case  of  all  is  that  in  which  tlie 
lines  are  at  right  angles  to  one  another. 

Let  the  force  R.,  given  to  be  resolved,  be  represented  by  MD,' 
and  let  the  angles  which  it  forms  with  the  lines  of  resolution  be 
given,  either  a,  (3,  y,  or  y,  ^.     Required  the  components  X,  Y,  Z. 

1°.     Suppose  a,  /?,  y  are  given,  then  we  deduce 
from  equation  (9)  X^  R  cos  a ; 

from  equation  (10)  Y ~  R  cos  /?; 

and  from  equation  (8)  Z=  R  cos  y. 

2°.  Suppose  the  data  are  R,  y,  <^,  that  is,  the  magnitude  of  the 
resultant,  its  inclination  to  one  of  the  axes  of  resolution,  and  the 
inclination  of  the  plane  of  the  resultant  and  that  axis  to  either  of  the 
other  axes. 

To  find  the  components  X  and  Y :  resolve  the  force  R  in  the 
vertical  plane  CMED  into  two  rectangular  components  along  MC 
and  ME.  Let  the  angle  CMD  be  denoted  by  y.  Then  we  have 
for  the  coiriponcnt  along  MC, 

Z=R  cosy,  i(ii) 

and  for  the  component  along  ME, 

ME  =  R  sin  y. 

Next,  resolve  the  component  along  ME  in  the  horizontal  plane 
BMAE,  into  two,  one  along. j^i^j-a^iid  the  other  along  MB.     Let 


STATICS  OF  A  PARTICLTT.— ATTRACTION:      165 

the  angle  EMA  be  denoted  by  4>.      Then  we  have  for  the   com- 
ponent along  MA^ 

X=  ME  cos  4>  =  Rs'm  y  cos  <^,  (12) 

and  for  the  component  along  MB, 

y=  J/£' sin  9  =i?  shi  y  sin  (^.  (13) 

469.  We  are  now  prepared  to  solve  the  general  problem: — Ci\en, 
any  number  of  forces  acting  on  one  point,  in  lines  which  lie  in 
different  planes,  required  their  resultant  in  position  and  magni- 
tude. 

Through  the  point  acted  on,  draw  three  lines  or  axes  of  resolution 
at  right  angles  to  one  another.  Resolve  each  force,  by  J^  468,  1°,  or 
by  §  468,  2°,  into  three  components,  acting  respectively  along  the 
three  lines.  When  all  the  forces  have  been  thus  treated,  add  severally 
the  sets  of  components:  by  this  means,  all  the  forces  are  reduced  to 
three  at  right  angles  to  one  another.  Find,  by  equation  (2),  their 
resultant:  the  single  force  thus  obtained  is  the  resultant  of  the  given 
forces,  which  was  to  be  found. 

Remark. — All  the  remarks  made  with  reference  to  the  resolulion 
and  composition  of  forces  along  two  axes  (>;  435)  apply,  with  the  ne- 
cessary extension,  to  that  of  forces  along  three. 

470.  We  are  now  prepared  to  answer  the  question  which  forms 
the  first  general  head  of  Statics;  What  air  the  conditions  of  Equi- 
librium of  a  materia! point "}  The  answer  may  be  put  in  one  or  other 
of  two  forms. 

i^  If  a  set  of  forces  acting  on  a  material  point  be  in  equilibrium, 
any  one  of  tliem  must  be  equal  and  opposite  to'  the  resultant  of  the 
others:  or, 

2".  If  a  set  of  forces  acting  on  a  material  point  be  in  equilibrium, 
tlie  resultant  of  the  wliole  set  inu.st  be  C(iua!  to  noilnng. 

471.  Let  us  consider  the  first  of  these  statements. 

Given,  a  set  of  forces,  J\^  E,^,  E.^,  Sec,  in  equilibrium:  the  foicc 
E^,  for  example,  is  equal  and  opposite  to  the  resultant  of  E,,,  E,,  ^c  ; 
or,  the  resultant  of  E^,  E^,  &c.,.  is  -/•,.  Omitting  I\,'  fnd  the 
resultant  of  the  remaining  forcL-s  by  the  general  method;  the  com- 
ponents of  this  resultant  will  be 

E^  cos  a,^  +  E3  cos  a,  4-  &c.  along  AfX. 
E,  cos  j8,  +  P3  cos  /?3  +  &c.  along  //K 
I\  cos  y,  +  E^  COS  y^  +  (S:c.  along  AfZ. 

Now,  if  -  Ej  be  the  resultant,  the  components  o(  ~  E  will  be 
equivalent  respectively  to  the  components  of  tliis  rcsulfcmt,  there- 
fore 

-  wP,  cos  a,  ~  E^  cos  a^  +  E^  COS  a^  4-  &C. 

-  E^  cos  /?,  =  I\  cos  /i^  +  I\  COS  p^  -I-  &c. 

-  E^  cos  y,  =  Ej  cos  y_,  +  P  £05  y,  -<-  >S:c 


1 66  ABSTRACT  DYNAMICS. 

Which  equations,  in  the  following  more  general  form,  express  the 
required  conditions: 

P^  cos  Oj  +  P^  cos  a^  +  P.^  cos  Og  +  &c.  =  o. 

/*,  cos  /?,  +  P^  cos  /^a  +  7*3  cos  ^^  +  &:c.  =  o. 

P^  cos  Yj  +  7^2  COS  y,  +  /"j  COS  73  +  &c.  =  o. 

472.  The  second  form  of  the  answer  may  be  illustrated  either  a, 
dynamically,  or  b^  algebraically. 

{a)  Suppose  all  the  forces  reduced  to  three,  X,  Y,Z,  acting  at  right 
angles  to  each  other.  Under  what  circumstances  will  three"  forces 
give  a  vanishing  resultant?  Substitute  for  X  and  y  their  resultant 
P',  and  consider  R'  and  Z  at  right  angles  to  one  another.  If  they 
give  a  vanishing  resultant,  that  is,  if  Z  and  P  balance,  they  must 
either  be  equal  and  directly  opposed,  or  else  they  must  each  be  equal 
to  nothing.  But  they  are  not  directly  opposed,  therefore  each  is 
equal  to  nothing.  Now,  since  R'=o,  X  and  Y,  which  are  equi- 
valent to  R',  must  also  each  be  equal  to  nothing:  in  order,  therefore, 
that  the  resultant  of  forces  acting  along  three  lines  at  right  angles  to 
one  another  may  vanish,  we  have 

P^  cos  a,  +  P^  cos  a,  +  &c.  =  o. 

Pi  cos  /3i  +  Pi  cos  f^l  +  &c.  =  o. 

Pi  cos  yi  +  P^  cos  y^  +  &c.  =  o. 

(Z-)   The  general  expression  for  the  resultant  is 

R'  =  X'  +  Y'-^Z'. 

Now,  for  equilibrium,  R-o, 

and  therefore,  X'  +  Y"-+Z'^o. 

But  the  sum  of  three  positive  quantities  can  be  equal  to  nothing,  only 
when  each  of  them  is  nothing  ;  hence 

X=o, 

y=o, 

Z=o. 

473.  We  may  take  one  or  two  particular  cases  as  examples  of  the 
general  results  above.     Thus, 

1.  If  the  particle  rest  on  a  sniooth  curve,  the  resolved  force  along 
the  curve  must  vanish. 

2.  If  the  curve  be  rough,  the  resultant  force  along  it  must  be 
balanced  by  the  friction. 

3.  If  the  particle  rest  on  a  smooth  surface,  the  resultant  of  the 
applied  forces  must  evidently  be  perpendicular  to  the  surface. 

4.  If  it  rest  on  a  rough  surface,  friction  will  be  called  into  play, 
resisting  motion  along  the  surface;  and  there  will  be  equilibrium  at 
any  point  within  a  certain  boundary,  determined  by  the  condition 
that  at  /V  the  friction  is  /x  times  the  normal  pressure  on 'the  surface, 
while  within  it  the  friction  bears  a  less  ratio  to  the  normal  pressure.' 
W]ien,the  only  applied  force  is  gravity,  we  have  a  very  simple  result, 
Which  is. often 'practically  useful.     Let  6  be  the  angle  between  the 


STATICS    OF  A   PARTICLE.— ATTRACTION'.      167 

normal  to  the  surface  and  the  vertical  at  anv  point ;  the  normal 
pressure  on  the  surface  is  evidently  /Kcos  B,  where  //'is  the  weight 
of  the  particle;  and  the  resolved  part  of  the  \vciqht  parallel  to  the 
surface,  which  must  of  course  be  balanced  by  the  friction,  is  /Fsin  Q. 
In  the  limiting  position,  when  sliding  is  just  about  to  commence,  the 
greatest  possible  amount  of  statical  friction  is  called  into  play,  and 
we  ha\e  11^ sin  0  =  fjL J}/ cos  6, 

or  tan  6  :^  fx. 

The  value  of  0  thus  found  is  called  the  Angk  of  Rf:posc,  and  n-:ay 
be  seen  in  nature  in  the  case  of  sand-heaps,  and  slopes  formed  by 
debris  from  a  disintegrating  cliff  (especially  of  a  flat  or  laminated 
character),  on  which  the  lines  of  greatest  slope  are  inclined  to  the 
horizon  at  an  angle  determined  by  this  consideration. 

474.  A  most  important  case  of  the  composition  of  forces  acting 
at  one  point  is  furnished  by  the  consideration  of  the  attraction  of  a 
body  of  any  form  upon  a  material  particle  anywhere  situated.  Plxperi- 
ment  has  shown  that  the  attraction  exerted  by  any  portion  of  matter 
upon  another  is  not  modified  by  the  neighbourhood,  or  even  by  the 
interposition,  of  other  matter;  and  thus  the  attraction  of  a  body  on  a 
])article  is  the  resultant  of  the  several  attractions  exerted  by  its  parts. 
To  treatises  on  applied  mathematics  we  must  refer  for  the  examina- 
tion of  the  consecjuences,  often  very  curious,  of  various  laws  of 
Attraction ;  but,  dealing  with  Natural  Philosophy,  we  confine  our^ 
selves  to  the  law  of  gravitation,  which,  indeed,  furnishes  us  with  an 
ample  supply  of  most  interesting  as  well  as  useful  results. 

475.  This  law,  which  (as  a  property  of  matter)  will  be  carefully 
considered  in  the  next  Division  of  this  Treatise,  may  be  trrtis 
enunciated. 

Every  pariide  of  7natter  in  the  zaiiverse  attracts  cicry  o/hcr farticu' 
1^'ith  a  fcrce,  ivhosc  direction  ii  t/iat  cf  iJie  line  joi)iin\^  the  /r^v,  and 
liliose  ma:^nitude.  is  directly  as  t  tie  product  cf  their  masses,  and  inz-crse/y 
as  the  square  of  their  distance  front  each  other. 

Experiment  shows  (as  will  be  seen  further  on)  that  the  same  law 
liolds  for  electric  and  magnetic  attractions  ;  and  it  is  probable  that  it 
is  the  fundamental  law  of  all  natural  actionj  at  least  when  the  acting 
bodies  are  not  in  actual  contact. 

476.  For  the  special  applications  of  Statical  principles  to  which 
we  proceed,  it  will  be  convenient  to  use  a  special  unit  of  mass,  or 
quantity  of  matter,  and  corresponding  units  for  the  measurement  of 
electricity  and  magnetism. 

Thus  if,  in  accordance  with  the  physical  law  enunciated  in  §  475, 
we  take  as  the  expression  for  the  forces  exerted  on  each  other  by 
masses  J/ and  m,  at  distajicc  X>,  the  quantity 

Mm  ^ 

it  is  obvious  that  our  unit  force  is  the  mutual  attraction  of  two  units 
of  mass  placed  at  unit  of  distance  from  each  other. 


i68  ABSTRACT  DYNAMICS. 

4tn.  It  is  convenient  for  many  applications  to  speak  of  ttie  <?^«j//j^ 
of  a  distribution  of  matter,  electricity,  etc.,  along  a  line,  over  a  sur- 
face, or  through  a  volume. 

Here  density  of  line        is  thequantity  of  matter  per  unit  of  length. 
,,  ,,         surface       „  „  „  „  area. 

,i  „        volume      „  „  „  „        volume. 

478.  In  applying  the  succeeding  investigations  to  electricity  or 
magnetism,  it  is  only  necessary  to  premise  that  M  and  7?i  stand  for 
quantities  of  free  electricity  or  magnetism,  whatever  these  may  be,  and 
that  here  the  idea  of  inass  as  depending  on  inertia  is  not  necessarily 

involved.     The  formula  -y^  will  still  represent  the  mutual  action,  if 

\\c  take  as  unit  of  imaginary  electric  or  magnetic  matter,  such  a  quan- 
tity as  exerts  unit  force  on  an  equal  quantity  at  unit  dA^nce.  Here, 
however,  one  or  both  of  M,  711  may  be  negative ;  and,  as  in  these 
applications  like  kinds  repel  each  other,  the  mutual  action  will  be 
attraction  or  repulsion,  according  as  its  sign  is  negative  or  positive. 
With  these  provisos,  the  following  theory  is  applicable  to  any  of  the 
above-mentioned  classes  of  forces.  We  commence  with  a  few  simple 
cases  which  can  be  completely  treated  by  means  of  elementary  geo- 
metr)'. 

479.  If  the  different  points  of  a  spherical  iurface  attract  equally 
with  forces  varying  inversely  as  the  squares  of  the  distances,  a  particle 
placed  within  the  surface  is  not  attracted  in  any  direction. 

Let  HIKL  be  the  spherical  surface,  and  P  the  particle  within  it. 
Let  two  lines  HK,  JL,  intercepting  very  small  arcs  ///,  KL,  be 
drawn  through  P;  then,  on  account  of  the 
similar  triangles  HPI,  KPL,  those  arcs  will 
be  proportional  to  the  distances  IIP,  LP; 
and  any  small  elements  of  the  spherical  sur- 
face atZT/and  KL,  each  bounded  all  round 
by  straight  lines  passing  through  P  [and  very 
nearly  coinciding  with  //AT],  will  be  in  the 
duplicate  ratio  of  those  lines.  Hence  the 
forces  exercised  by  the  matter  of  these  ele- 
ments on  the  particle  P  are  equal ;  for  they 
are  as  the  quantities  of  matter  directly,  and  the  squares  of  the 
distances,  inversely;  and  these  two  ratios  compounded  give  that  of 
equality.  The  attractions  therefore,  being  equal  and  opposite,  de- 
stroy one  another:  and  a  similar  proof  shows  that  all  the  attractions 
due  to  the  whole  spherical  surface  are  destroyed  by  contrary  attrac- 
tions. Hence  the' particle  P  is  not  urged  in  any  direction  by  these 
attractions. 

480.  The  division  of  a  spherical  surface  into  infinitely  small  ele- 
ments, will  frccjucntly  occur  in  the  investigations  which  follow:  and 
Newton's  method,  described  in  the  preceding  demonstration,  in  which 
tlie  division  is  effected  in  sucli  a  manner  that  all  the  parts  may  be 
taken  together  in  pairs  of  opposite  elements  with  reference  to  an  internal 


STATICS  OF  A  PARTICLE.— ATTRACTION,      lOj 

point ;  besides  other  methods  deduced  from  it,  suitable  to  the  special 
problems  to  be  examined;  will  be  repeatedly  employed.  The  follow- 
ing digression  (§.§  4S1,  486),  in  which  some  definitions  and  elemen- 
tary geometrical  propositions  regarding  this  subject  are  laid  down, 
will  simplify  the  subsequent  demonstrations,  both  by  enabling  us, 
through  the  use  of  convenient  terms,  to  avoid  circumlocution,  and 
by  affording  us  convenient  means  of  reference  for  elementary  prin- 
ciples, regarding  which  repeated  explanations  might  otherwise  be 
necessary. 

481.  If  a  straight* line  which  constantly  passes  through  a  fixed 
point  be  moved  in  any  manner,  it  is  said  to, describe,  or  generate, 
a  conical  surface  of  which  the  fixed  point  is  the  vertex. 

If  the  generating  line  be  carried  from  a  given  position  continuously 
through  any  series  of  positions,  no  two  of  which  coincide,  till  it  is 
brought  back  to  the  first,  the  entire  line  on  the  two  sides  of  the  fixed 
point  will  generate  a  complete  conical  surface,  consisting  of  two 
sheets,  which  are  called  vertical  or  opposite  cones.  Thus  the  elements 
HI  '\x\<\  KL,  described  in  Newton's  demonstration  given  above,  may 
be  considered  as  being  cut  from  the  spherical  surfiice  by  two  opposite 
cones  having  T  for  their  common  vertex. 

482.  If  any  number  of  spheres  be  described  from  the  vertex  of  a 
cone  as  centre,  the  segments  cut  from  the  concentric  spherical  sur- 
faces will  be  similar,  and  their  areas  will  be  as  the  squares  of  the 
radii.  The  quotient  obtained  by  dividing  the  area  of  one  of  these 
segments  by  the  square  of  the  radius  of  the  spherical  surface  from 
which  it  is  cut,  is  taken  as  the  measure  of  the  sdid  angle  of  the  cone. 
The  segments  of  the  same  splierical  surlaces  made  by  the  opposite 
cone,  are  respectively  equal  and  siniilar  to  the  former.  Hence  the 
solid  angles  of  two  vertical  or  opposite  cones  are  equal :  either  may 
be  taken  as  the  solid  angle  of  tlie  complete  conical  surtace,  of  wliicl^ 
the  opposite  cones  are  the  two  sheets. 

483.  Since  the  area  of  a  spherical  surfice  is  equal  to  the  s<juare  of 
its  radius  multiplied  by  4^:,  it  follows  that  the  sum  of  tlie  suliJ  angles 
of  all  the  distinct  cones  which  can  be  described  with  a  given  point  as 
vertex,  is  equal  to  4- 

484.  The  solid  angles  of  vertical  or  opposite  cones  being  equal, 
we  may  infer  from  what  precedes  that  the  sum  of  the  solid  angles 
of  all  the  complete  conical  surfaces  which  can  be  described  with. 
out  mutual  intersection,   with   a  given   point   as  vertex,   is   equ<.tl 

to   2~. 

485.  Tlie  solid  angle  subtended  at  a  point  by  a  superficial  area  of 
any  kind,  is  the  solid  angle  of  the  cone  generated  by  a  straigli.t  line 
passing  through  the  point,  and  carried  entirely  round  the  boundary 
of  the  area. 

48G.  A  very  small  cone,  that  is,  a  cone  such  that  any  two  posi- 
tions of  the  generating  line  contain  but  a  very  small  angle,  is  sa;.l  to 
be  cut  at  right  angles,  or  oiJiogonally,  by  a  spherical  surfice  de- 


1 7  o  ABSTRA  CT .  DYNAMICS. 

scribed  from  its  vertex  as  centre,  or  by  any  surface,  whether  plane  or 
curved,  which  touches  the  spherical  surface  at  the  part  where  the  cone 
is  cut  by  it. 

A  very  small  cone  is  said  to  be  cut  obliquely,  when  the  section  is 
inclined  at  any  finite  angle  to  an  orthogonal  section;  and  this  angle 
of  inclination  is  called  the  obl'ujuity  of  the  section. 

The  area  of  an  orthogonal  section  of  a  very  small  cone  is  equal  to 
the  area  of  an  oblique  section  in  the  same  position,  multiplied  by  the 
cosine  of  the  obliquity. 

Hence-  the  area  of  an  oblique  section  of  a  small  cone  is  equal  ta 
the  quotient  obtained  by  dividing  the  product  of  the  square  of  its 
distance  from  the  vertex,  into  the  solid  angle,  by  the  cosine  of  the 
obliquity. 

487.  Let  E  denote  the  area  of  a  very  small  element  of  a  spherical 
surface  at. the  point  E  (that  is  to  say,  an  element  every  part  of  which 
is  very  near  the  point  E),  let  w  denote  the  solid  angle  subtended  by  E 
at  any  point  E,  and  let  EE,  produced  if  necessary,  meet  the  surface 
again  in  E' ;  -then  a  denoting  the  radius  of  the  spherical  surface,  we 
have 

2a.io.EE^ 
^~~~E£^~' 

For,  the  obliquity  of  the  element  E,  considered  as  a  section  of  the 
cone  of  which./*  is  the  vertex  and  the  element 
E  a  section  (being  the  angle  between  the 
given  spherical  surface  and  another  described 
from  E  as  centre,  with  EE  as  radius),  is- 
equal  to  the  angle  between  the  radii  EE  and 
EC,  of  the  two  spheres.  Hence,  by  con- 
sidering the  isosceles  triangle  ECE,  we  find 
that  the  cosine  of  the  obliquity  is  equal  to 

^EE'  EE' 

'—frpr  or  to  , 

EC  2a  ^ 

and  we  arrive  at  the  preceding  expression  for  E. 

488.  T/ie  attraction  of  a  luiiform  spherical  surface  on  an  cxtcrnaT 
point  is  the  same  as  if  tJic  .ivJwlc  jnass  were  collected  at  the  centre^. . 

Let  E  be  the  external  point,  C  the  centre  of  the  sphere,  and  CAE 
a  straight  line  cutting  the  spherical  surface  in  A.  Take  /  in  C/V 
so  that   CE,  CA,  C/ may  be  continual  proportionals,  and   let.  the 

^  This  theorem,  which  is  more  comprehensive  than  that  of  Newton  in  his  first 
proposition  regarding  attraction  on  an  external  point  (Prop.  LXXI.),- i.s  fully  es- 
tablished as  a  corollary  to  a  subsequent  proposition  (LXXIII.  cor.  i).  If  we  had 
considered  the  proportion  of  the  forces  exerted  upon  two  external  points  at 
(Hfferent  distances,  instead  of,  as  in  the  "text,  investigating  the  absolute  force  on? 
one  point,  and  if  besil^es  we  had  taken  together  all  the  pairs  of  elements  which 
would  constitute  two  narrow  annular  portions  of  the  surface,  in  planes  perpen- 
dicular to- /'C,  the  theorem  and  its  dcmonstr-^.tion  \,'ould  have  coincided  precisely 
with  Prop?  LXXI.  of  the  rriiuipia. 


STATICS   OF  A  PARTICLE.— ATTRACTIO. 


»7» 


whole  spherical '  surface  be  di- 
vided into  pairs  of  opposite  ele- 
vients  zc>it/i  reference  io  the  point 
I. 

Let  H  and  //'  denote  the 
magnitudes  of  a  pair  of  such 
elements,  situated  respectively 
at  the  extremities  of  a  chord 
HH' ;  and  let  cj  denote  the 
magnitude  of  the  solid  angle 
subtended  by  either  of  these  elements  at  the  point  I. 

V.'e  have  (§  4S6), 

,,   ^.iir      ,_    .^.iH" 

//=  T^TTT?)  and  H  = ,  ■  . 

COS  C///  cosCH  I 

Hence,  if  p  denote  the  density  of  the  surface,  the  attractions  of  the 
two  elements  7/  and  //'  on  F  are  respectively 


IIF 


,  and  p  • 


iir 


^  cos  CHI  •  Fir '  "''"  ^  cos  CHI '  FH* ' 
Now  the  two  triangles  FCII.  IICI  have  a  common  angle  at  C,  and, 
since  FC  :  CH ::  CII :  CI,  the  sides  about  this  angle  are  propor- 
tional. Hence  the  triangles  are  similar;  so  that  the  angles  CFH 
and  CHI  2.XQ.  equal,  and 

LII  __CII_a_ 

YlF~  CF  ~  CF' 
In  the  same  way  it   may  be   proved,  by  considering   the    triangles 
FCIF,  11  CI,  that  the  angles  CFH  and  CIIId.1^  eciual,  and  that 

//r  _  CIF  _ _a 

'lfF~  CF  ~  CF' 
Hence  the  expressions  for  the  attractions  of  the  elements  //and  IF 
on  F  become 

or  <Z  ,  (I)  a 

^co^  CHI'  CF''  ^^^cos  CHF'  CF' ' 
which  are  equal,  since  the  triangle  IICIF  is  isosceles;  and,  for  the 
same  reason,  the  angles  CFH,  CFH',  which  have  been  proved  to 
be  respectively  equal  to  the  angles  CHI,  CHF,  are  equal.  We 
infer  that  the  resultant  of  the  forces  due  to  the  two  elements  is  in 
the  direction  PC,  and  is  equal  to 

To  find  the  total  force  on  F,  we  must  take  the  sum  of  all  the 
forces  along /'C  due  to  the  pairs  of  opposite  elements;  and,  since 
the  multipUer  of  a>  is  the  same  for  each  pair,  we  must  add  all  the 
values  of  w,  and  we  therefore  obtain  (§  483),  for  the  required  re- 
sultant, 

^CF*  * 


,  7  2  ABSTRA  CT  D  YNAMICS. 

The  numerator  of  this  expression  (being  the  product  6f  the'density 

into  the  area  of  the  spherical  surface)  is  equal  to  the  mass  of  the 

entire  charge;  and  therefore  the  force  on  F  is  the  same  as  if  the 

whole  mass  were  collected  at  C. 

Cor.     The  force  on  an  external  point,  infinitely  near  the  surface, 

is  equal  to  47rp,  and  is  in  the  direction  of  a  normal  at  the  point. 

The  force  on  an  internal  point,  uowever  near  the  surface,  is,  by 

a  preceding  proposition,  equal  to  nothing. 

489.  "  Let  (T  be  the  area  of  an  infinitely  small  element  of  the  surface 
at  any  point  Py  and  at  any  other  point  JI  of 
the  surface  let  a  small  element  subtending  2^ 
solid  angle  w,  at  jP,  be  taken.  The  area  of 
this  element  will  be  equal  to 


cos  CHF' 

and  therefore  the  attraction  along  HP,  which 

it  exerts  on  the  element  o-  at  i^  will  be  equal 

to  poi.  p<r  ill  J 

cos  CHF'   °^  cos  CJIP''  '^' 

Now  the  total  attraction  on  the  element  at  P  is  in  the  direction  CP; 
the  component  in  this  direction  of  the  attraction  due  to  the  element 
JI,is 

<i) .  p  V ; 
and,  since  all  the  cones  corresponding  to  the  different  elements  of  the 
spherical  surface  lie  on  the  same  side  of  tlie  tangent  plane  at  P,  we 
deduce,  for  the  resultant  attraction  on  the  element  cr, 

27rpV. 

From  the  corollary  to  the  preceding  proposition,  it  follows  that  this 
attraction  is  half  the  force  which  would  be  exerted  on  an  external 
point,  possessing  the  same  quantity  of  matter  as  the  element  <r,  and 
placed  infinitely  near  the  surface. 

490.  In  some  of  the  most  important  elementary  problems  of  the 
theory  of  electricity^  spherical  surfaces  with  densities  varying  inversely 
as  the  cubes  of  distances  from  excentric  points  occur:  and  it  is  of 
fundamental  importance  to  find  the  attraction  of  such  a  shell,  on  an 
internal  or  external  point.  This  may  be  done  synthetically  as  follov/s; 
the  investigation  being,  as  we  shall  see  below,  virtually  the  same 
as  that  of  §  479,  or  §  488. 

491.  Let  us  first  consider" the  case  in  which  the  given  point  .S  and 
the  attracted  point  P  are  separated  by  the  spherical  surface.  The 
two  figures  represent  the  varieties  of  this  case  in  which,  the  point  JS" 
being  v.ithout  the  sphere,  .P  is  within;  and,  S  being  within,  the 
attracted  point  is  external.  The  same  demonstration  is  applicable 
literally  with  reference  to  the  two  figures;  but,  for  avoiding  the  cor^- 
sideration  of  negative  quantities,  some  of  the  expressions  may  be 
conveniently  modified  to  suit  the  second  figure.    In  such  instances 


STATICS  OF  A  PARTICLE.— ATTRACTION.      173 


the  two  expressions  are  given  in  a  double  line,  the  upper  being  that 
which  is  most  convenient  for  the  first  figure,  and  tlie  lower  for  the 
second. 

l,ct  the  radius  of  the  sphere  be  denoted  by  a,  and  let  /  be  the 
distance  of  S  from  C,  the  centre  of  the  sphere  (not  represented  in 
the  figures). 

Join  iS/'and  take  Tm  this  hne  (or  its  continuation)  so  that 
(fig.  i)     SP.ST=r-a\ 
(fig.  2)     SP.TS=a'-f\ 
Through  T  draw  any  line  cutting  the  spherical  surface  at  K,  K'. 
Join  SK.,  SK\  and  let  the  lines  so  drawn  cut  the  spherical  surface 
again  in  E,  E. 

Let  the  whole  spherical  surface  be  divided  into  pairs  of  opposite 
elements  with  reference  to  the  point  T.  Let  jTand  K'  be  a  pair  of 
such  elements  situated  at  the  extremities  of  the  chord  KK,  and 
subtending  the  solid  angle  w  at  the  point  T ;  and  let  elements  E  and 
E'  be  taken  subtending  at  5  the  same^  solid  angles  respectively  as  the 
elements  K  and  K'.  By  this  means  we  may  divide  the  whole 
spherical  surface  into  pairs  of  conjugate  elements,  E,  ^"V  since  it  is 
easily  seen  that  when  we  have  taken  every  pair  of  elements,  A',  A", 


the  whole  surface  will  have  been  exhausted,  without  repetition,  by  tlic 
deduced  elements,  E,  E'.  Hence  the  attraction  on  P  will  be  the 
final  resultant  of  the  attractions  of  all  the  pairs  of  elements,  E,  E. 

Now  if  p  be  the  surface  density  at  E,  and  if  jp  denote  the  attraction 
of  the  element  E  on  P,  we  have 

^     EP' 
According  to  the  given  law  of  density  we  shall  have 

\ 


P  = 


SE' 


where  \  is  a  constant.     Again,  smce  SEKh  equally  inclined  to  the 
spherical  surface  r\^  the  two  points  of  intersection,  we  have 


and  hence 


F= 


SE'' 


E  = 

SE' 
SK*' 


SE' 
SK' 


SE' 


laui.TK' 


'.a ID.  TK 


EP* 


-^•AX' 


TK 


SE.SK'.EP' 


174  ABSTRACT  DYNAMICS. 

Now,  by  considering  the  great  circle  in  which  the  sphere  is  cut  by  tt' 
plane  through  the  line  SK,  we  find  that 

(fig.  i)     SK,SE=r-a\ 

(fig.  2)     KS.SE  =  a^-f\ 
and  hence  SK.  SE  =  SP .  ST,  from  which  we  infer  that  the  triangles 
KST,  FSE  are  similar;  so  that  TK .  SK  w  PE  \  SP.     Hence 
TK^  I 


SK\PE*     SP 


)i  > 


and  the  expression  for  i^  becomes 

P_.    J^_  __^ 

^~      KIC  SE.SP^'"^' 

Modifying  this  by  preceding  expressions  we  have 

(fig.  I)  F=\. -^, . ^^^, _^.j^s .  SK, 

(fig.  2)  p^x. m-^-(^~/^^-sP^'^^' 

iSimilarly,  if  P'  denote  the  attraction  of  .£'  on  P,  we  have 
(fig.  i)   ir'^xJ^,.^__^-_^.5ir', 

(fig.    2)     i^'  -  X  ^^  .  ^^2  _jr^)SP'  ■  ^'"^^ 

Now  in  the  triangles  which  have  been  shown  to  be  similar,  the 
.mgles  TA'S,  EPS  are  equal;  and  the  same  may  be  proved  of  tlie 
angles  K'ST,  PSE'.  Hence  the  tv\-o  sides  ^A',  ^A"'  of  the  triangle 
KSK'  are  mch'ned  to  the  third  at  the  same  angles  as  those  between 
the  line  PS  and  directions  PE,  PE'  of  the  two  forces  on  the  point 
P ,  and  the  sides  SK,  SK'  are  to  on-e  another  as  the  forces,  F,  F', 
m  the  directions  PE,  PE'.  It  follovvs,  by  '  the  triangle  of  forces,' 
that  the  resultant  of  7^ and  F'  is  along  PS,  and  that  it  bears  to  the 
component  forces  the  same  ratios  as  the  side  A'A''  of  the  triangle 
bears  to  the  other  two  sides.  Hence  the  resultant  force  due  to  the. 
two  elements  E  and  E'  on  the  point  P,  is  towards  S,  ond  is  equal  to 

V       2a  fc)  \,2/Z ,  tir 

■  Kjc'  (p'-^'jTsp'      '  °^  {r'-^iYsy^ ' 

The  total  resultant  force  will  consequently  be  towards  S'^  and  we 
find,  by  summation  (§  4G6)  for  its  magnitude; 

lr-a^)SP'' 

Hence  we  infer  that  the  resultant  force  at  any  point  P,  separated 
from  ^  by  the   spherical  surface,  is  the  same  as  if  a  quantity  of 

matter  equal  to  ->-j— ^  were  concentrated  at  the  point .?. 


STATICS  OF  A  PARTICLE.— ATTRACTION.      175 

492.     To  find  the  attraction  when  5  and  P  are  cither  both  without 
or  both  witliin  the  spherical  suri'ace. 

Take  in  CS^  or  in  CS  produced  through  6",  a  point  S^,  such  that 

CS.  CS^-^a". 

Then,  by  a  v/ell-known  geometrical  theorem,  if  E  be  any  point  on 
the  spherical  surface,  we  have 

SE  ^f 
S,E      a  * 


Hence  v;e  have 


\a^ 


SE'     f\S^E'' 
Hence,  p  being  the  electrical  density  at  E,  we  have 

^      S^E'      S,E'' 

f 

Hence,  by  the  Investigation  in  the  preceding  section,  the  attraction 
on  P  is  towards  S^^  and  is  the  same  as  if  a  quantity  of  matter  equal 
^    Xi .  477-a  ,       , 

to  -ji J  were  concentrated  at  that  pomt  j 


Xi 


^'1 


/,  being  taken  to  denote  C5,.     If  for  f^  and  \  we  substitute  their 

a*  X/z' 

values,  -J  and  -^5- ,  we  have  the  modified  expression 


for  the  quantity  of  matter  which  we  must  conceive  to  be  collected 
at^. 

403.  If  a  spherical  surface  be  electrified  in  such  a  way  that  the 
electrical  density  varies  inversely  as  the  cube  of  the  distance  from 
an  internal  point  S,  or  from  the  corresponding  extenial  pomt  S-^^  it 
will  attract  any  external  point,  as  if  its  whole  electricity  were  con- 


176  .  ABSTRA  CT  D  YNAMICS. 

centrated  at  S,  and  any  internal  point,  as  if  a  quantity  of  electricity 
greater  than  its  own  in  the  ratio  of  a  to /were  concentrated  at  ^ 

Let  the  density  at  E  be  denoted,  as  before,  by  -^3 .    Then,  if  we 

consider  two  opposite  elements  at  E  and  E\  which  subtend  a  soHd 

angle  w  at  the  point  S,  the  areas  of  these  elements  being    '■      '    — 

ui,2a.SE" 
and  — ■'  „  ^> '  ^^^  quantity  of  electricity  which  they  possess  will  be 


£E' 

\  .  2a  .0)/    I  I    \  X.  2fl  . 

~1^£^  [SE  "^  SE'J  °^  SETS 


SE .  SE  ' 

Now  SE .  SE'  is  constant  (Euc.  III.  35)  and  its  value  Is  a*-/*. 
Hence,  by  summation,  we  find  for  the  total  value  of  electricity  on 
the  spherical  surface 

X .  4Tra 

Hence,  if  this  be  denoted  by  m,  the  expressions  in  the  preceding 
paragraphs,  for  the  quantities  of  electricity  which  we  must  suppose  to 
be  concentrated  at  the  point  6"  or  S^,  according  as  /*  is  without  or 
within  the  spherical  surface,  become  respectively 

ni,  and  ^  m, 

494.  The  dinrf  analyiical  solution  of  such  problems  consists  in 
the  expression,  by  §  408,  of  the  three  components  of  the  whole  at- 
traction as  the  sums  of  its  separate  parts  due  to  the  several  particles 
of  the  attracting  body  3  the  transformation,  by  the  usual  methods,  of 
tliese  sums  into  deiimte  integrals;  and  the  evaluation  of  the  latter. 
This  is,  in  general,  inferior  in<  elegance  and  simplicity  to  the  less 
du-ect  mode  of  solution  depending  upon  the  determination  of  the 
potential  energy  of  the  attracted  partrde  with  reference  to  the  forces 
exerted  upon  it  by  the  attractmg  body,  a  method  which  we  shall 
presendy  develop  with  peculiar  care,  as  it  is  of  incalculable  value  in 
the  theories  of  Electricity  and  Magnetism  as  well  as  in  that  of 
Gravitation.  But  before  we  proceed  to  it,  we  give  some  instances  of 
the  direct  method. 

(a)  A  useful  case  is  that  of  the  attraction  of  a  circular  plate  of 
uniform  surface  density  on  a  point  in  a  .line  through  its  centre,  and 
perpendicular  to  its  plane. 

All  parallel  slices,  of  equal  thickness,  of  any  cone  attract  equally 
(both  in  m.agnitude  and  direction)  a  particle  at  the  vertex. 

For  the  proposition  is  true  of  a  cone  of  infinitely  small  ang'!e,  the 
masses  of  the  slices  being  evidently  as  the  squares  of  their  distances 
from  the  vertex-  If  /  be  the  thickness,  p  the  volume  density,  and  m 
the  angle,  the  attraction  is  w/p. 

All  slices  of  a  cone  of  infinitely  small  angle,  if  of  equal  thickness 


STATICS  OF  A   PARTICLE— ATTR ACTIO K     177 


and  equally  inclined  to  the  axis  of  the  cone,  exert  equal  forces  on  a 
particle  at  the  vertex.  For  the  area  of  any  inclined  section,  whatever 
be  Its  orientation,  is  greater  than  that  of  th.;  corresponding  transverse 
section  in  the  ratio  of  unity  to  the  cosine  of  the  angle  of  inclination. 

Hence  if  a  plane  touch  a  sphere  at  a 
point  B,  and  if  the  plane  and  sphere  have 
equal  surface  density  at  corresponding 
points  P  and  />  in  a  line  drawn  through 
A,  the  point  diametrically  opposite  to  B, 
correspondi.ng  elements  at  P  and  p  exert 
equal  attraction  on  a  particle  at  A. 

Thus  the  attraction  on  A,  of  any  part 
of  the  plane,  is  the  same  as  that  of  the 
corresponding  part  of  the  sphere,  cut  out 
by  a  cone  of  infinitely  small  angle  whose  vertex  is  A. 

Hence  if  we  resolve  along  the  line  AB  the  attraction  of /^  on  A, 
the  component  is  equal  to  the  attraction  along  Ap  of  the  transverse 
section  /r,  i.e.  /juj,  where  tu  is  the  angle  subtended  at  A  by  the 
element/^,  and  p  the  surface  density. 

Thus  any  portion  whatever  of  the  sphere  attracts  A  along  AB 
^-ith  a  force  proportional  to  its  spherical  opening  as  seen  from  A; 
and  the  same  is,  by  what  was  proved  above,  true  of  a  flat  plate. 

Hence  as  a  disc  of  radius  a  subtends  at  a  point  distant  h  from  it, 
in  the  direction  of  the  axis  of  the  disc,  a  solid  angle 

the  attraction  of  such  a  disc  is 


S'irp 


v>hich  for  an  infinite  disc  becomes,  whatever  the  distance  //, 

2irp. 

From  the  preceding  formula  many  useful  results  may  easily  be 
deduced  :  thus, 

{h)  A  uni'rrm  cylinder  of  length  /,  and  radius  a,  attracts  a  point 
in  its  axis  at  a  distance  x  from  the  r^earest  end  with  a  force 

V,nicn   the   cylinder  is  of  infinite  length  (in  one  direction)  the  at- 
traction IS  theretbre 

27rp  [Jx^  +0'  -x)', 

and,  when  the  attracted  particle  is  in  contact  with  the  centre  of  the 
end  oi  the  infinite  cylinder,  this  is 

2TTpa. 
{c)  A  right  cone,  of  semiveitical  angle  a,  and  length  /,  attracts  a 


1 7  8  ABSTRA  CT  D  YNAMICS. 

particle  at  its  vertex.     Here  we  have  at  once  for  the  attraction,  the 
expression 

27r/)/(l  -COS  a), 

which  is  simply  proportional  to  the  length  of  the  axis. 

It  is  of  course  easy,  when  required,  to  find  the  necessarily  less 
simple  expression  for  the  attraction  on  any  point  of  the  axis. 

(d)  For  magnetic  and  electro-magnetic  applications  a  very  useful 
case  is  that  of  two  equal  uniform  discs,  each  perpendicular  to  the  line 
joining  their  centres,  on  any  point  in  that  line — their  masses  (§  478) 
being  of  opposite  sign — that  is,  one  rcpeUing  and  the  other  attracting. 

Let  a  be  the  radius,  p  the  mass  of  a  superficial  unit,  of  eilhei,  c 
their  distance,  x  the  distance  of  the  attracted  point  from  the  nearest 
disc.     The  whole  force  is  evidently 


t  X  +  C  X         \ 


In  the  particular  case  v,-hen  r  is  diminished  without  limit,  this 
becomes 

a' 

{x"--^a')i 

495.  Lt*  P  and  P"  be  two  points  infinitely  near  one  another  on 
two  sides  of  a  surface  ever  which  matter  is  distributed ;  and  let  p  be 
the  density  of  tins  distribution  on  the  surface  m  the  neighbourhood 
of  these  points.  Then  whatever  be  the  resultant  attraction,  y?,  at  P, 
due  to  all  the  attracting  matter,  whether  lodging  on  this  surface,  or 
elsewhere,  the  resultant  force,  7?',  on  P'  is  the  resultant  of  a  force 
equal  and  parallel  to  7?,.  and  a  force  equal  to  A-n-p,  in  the  direction 
from  P  perpendicularly  towards  the  surface.  For,  suppose  PP'  to 
be  perpendicular  to  the  surface,  which  will  not  limit  the  generality  of 
the  proposition,  and  consider  a  circular  disc,  of  the  surface,  having 
its  centre  in  /IP',  and  radius  infinitely  small  in  comparison  with  the 
radii  of  curvature  of  the  surface  but  infinitely  great  in  comparison 
witli  PP.  This  disc  Will  [^  494]  attract  P  and  P  with  forces, 
each  equal  to  2Trp  and  opposite  to  one  another  in  the  line  PP'. 
AV'lience  t!ie  proposition.  It  is  one  of  much  importance  in  the  theory 
of  electricity. 

496.  It  may  be  shown  that  at  the  southern  base  of  a  hemispherical 
hill  of  radius  a  and  density  p.  the  true  latitude  (as  measured  by  the 
aid  of  the  plumb-line,  or  by  reflection  of  siarhght  in  a  trough  of 
mercury)  is  diminished  by  the  attraction  of  ilie  mountain  by  the 
angle 

"^p-ra 


G-*pa' 
where  G  is  the  attraction  of  the  earth,  estimated  in  the  same  units. 


STATICS   OF  A   PARTICLE.— ATTRACTION     179 

Hence,  if  ^  be  the  radius  and  o-  the  mean  density  of  the  earth,  the 
angle  is 

r^^l^^.  or  ^^approximately. 

Hence  the  latitudes  .of  stations  at  the  base  of  the  hill,  north  and 

south  of  it,  differ  by   ^  ( 2  +  —  ] ;  instead  of  by  -jr,  as  they  would 

do  if  the  hill  were  removed. 

In  the  same  way  the  latitude  of  a  place  at  the  southern  edge  of  a 

hemispherical  cavity  is  increased  on  account  of  the  cavity  by  \  —j^ 

where  p  is  the  density  of  the  superficial  strata. 

497.-    As  a  curious  additional  example  of  the  class  of  questions 
■  we  have  just  considered,  a  deep  crevasse,  extending  east  and  west, 
increases   the  latitude   of  places  at  its  southern  edge  by  (approx- 
imately) the  angle  -j  —^  where  p  is  tlie  density  of  the  crust  of  the 
earth,  and  a  is  the  width  of  the  crevasse.     Thus  the  north  'edge  of 

the  crevasse  will  have  a  lou.'cr  latitude  than  the  south  edge  if  5  -  >  i, 

'  <j      ' 

which  might  be  the  case,  as  there  are  rocks  of  density  1x5-5  or 

3*67  times  that  of  water.    At  a  considerable  depth  in  the  crevasse, 

this  change  of  latitudes  is  nearly  doubled,  and  then  the  southern  side 

has  the  greater  latitude  if  the  density  of  the  crust  be  not  less  than 

yS2>  times  that  of  waten 

498.  It  is  interesting,  and  will  be  useful  later,  to  consider  as  a 
particular  case,  the  attraction  of  a  sphere  whose  mass  iS'  composed  of 
concentric  layers,  each  of  uniform  density.  Let  a  be,  as  above,  the 
mean  density  of  the  whole  globe,  and  t  the  density  of  the  upper  crust. 

The  attraction  at  a  depth  //,  small  compared  wth  the  radius,  is 

^n<r,{R-h)=^G, 

where  cr,  denotes  the  mean  density  of  the  nucleus  remaining  when  a 
shell  of  thickness  h  is  removed  from  the  sphere.     Also,  evidently, 

■^Tra,  {R  -  hy  +  ^Tr{R-  Kfh  =  ^to-T?', 

or  G^  {R  -  hy  +  47rT  (/?  -  hy/i  -  OR", 

whence  G^  =  G  ii  +  -jz\  —  4117/1. 

The  attraction  is  therefore  unaltered  at  a  depth  h  if 
G    ,  .  . 

—  -  ^TTcr  =  27rT,    I.e.    T  =  f<T. 

499.  Some  other  simple  cases  may  be  added  here,  as  their  resuUs 
will  be  of  use  to  us  subsequently. 


i8o 


ABSTRACT  DYNAMICS. 


(a)  The  attraction  of  a  circular  arc,  AB,  of  uniform  density,  on  a 
particle  at  the  centre,  C,  of  the  circle,  lies 
evidently  in  the  line  C£>  bisecting  the  arc. 
Also  the  resolved  part  parallel  to  CD  of 
the  attraction  of  an  element  at  P  is 

mass  of  element  at  jP  n^r^ 
^^- COS..  i'CZ?. 

Now  suppose  the  density  of  the  chord  AB 
to  be  the  same  as  that  of  the  arc.  Then 
for  (mass  of  element  at  P>^cos  l  FCD) 
we  m,ay  put  (mass  of  projection  of  element 
on  AB  at  0;  since,  if  FT  be  the  tan- 
gent at  F,  L  FTQ  =  L  FCD. 


Hence  attraction  along  CD- 


sum  of  projected  elements 
CD" 


_pAB 
"CD"* 

if  p  be  the  density  of  the  given  arc, 

2p%\nj.  A  CD 
"  CD         ' 


It  is  therefore  the  same  as  the  attraction  of  a  mass  equal  to  th& 
chord,  with  the  arc's  density,  concentrated  at  the  point  D. 

{b)  Again,  a. limited  straight  hne  of  uniform  density  attracts  any- 
external  point  in  the 
same  direction  and  with 
tlie  same  force  as  the 
corresponding  arc  of  a 
circle  of  the  same  den- 
sity, which  has  the  point 
for  centre,  and  touches 
the  straight  line. 

For  if  CpF  be  drawn 
cutting  the  circle  in  p 
and  the  line  in  F,  ele- 


ment at  /  -:  element  sx  F  ::  Cp 


CP 

CP  ^'~  ;  that  is,  as  Cp' 


CP\ 


Hence  the  attractions  of  these  elements  on  C  are  equal  and  in  tlie 
same  line.  Thus  the  arc  ah  attracts  C  as  the  line  AB  does;  aix],  liy 
the  last  proposition,  the  attraction  of  ^j5  bisects  the  angle  y/C/-',  and 
is  equal  to 

-^^sin^.^C^. 


(STATICS  OF  A   PARTICLE.— ATTRACTION:     i8r 


{c)  This  may  be 
put  into  other  use- 
ful forms — thus,  let 
CKF  bisect  the  an- 
gle ACB,  and  let 
Aa,  Bb,  EF,  be 
drawn  perpendicular 
to  CF  from  the 
ends  and  middle 
point  of  A£.  We 
have 


-.^_    KB  .        _._         AB      CD 
sm  z  AC^  =  ^sm  .  CAD^-j^-^^^^~^. 


z  (AC+  CB)  {AC+  CB'-AB") 


CF. 


(0. 


Hence  the  attraction,  which  is  along  CK,  is 

2pAB  pAB 

{A  C  ^CB)  CK  " 

For  evidently, 

bK  :  Ka  ::  BK  :  KA  ::  BC  ■  CA  ::  bC  :  Ca, 

i.e.,  ab  is  divided,  externally  in  C,  and  internally  in  K,  in  the  same 
ratio.     Hence,  by  geometry, 


A'C.  CF=aC.  Cb  =  \{AC-^  CB'-AB'], 

which  gives  the  transformation  in  (i). 

{d)  CF  is  obviously  the  tangent  at  C  to  a  hyperbola,  passing 
through  that  point,  and  having  A  and  B  as  foci.  Hence,  if  in 
any  plane  through  AB  any  hyperbola  be  described,  with  foci  A 
and  B,  it  will  be  a  line  of  force  as  regards  the  attraction  of  the 
line  AB ;  that  i<:,  as  wll  be  more  fully  explained  later,  a  curve  which 
at  every  point  indioites  the  direction  of  attraction. 

(c)  Similarly,  if  a  prolate  spheroid  be  described  with  foci  A  and  B, 
and  passing  through  C,  CT^will  evidently  be  the  normal  at  C ;  thus 
the  force  on  a  particle  at  C  will  be  perpendicular  to  the  spheroid : 
and  the  particle  would  evidenlly  rest  in  equilibrium  on  the  surface, 
even  if  it  were  smooth.  This  is  an  instance  of  (what  we  shall  pre- 
sently develop  at  some  length}  a  surface  of  equilibrium,  a  level 
surface,  or  an  equipotential  surface. 

(/)  \Vc  may  further  y^rove,  by  a  simple  application  of  the 
preceding  theorem,  that  the  lines  of  force  due  to  the  attraction 
of  two  infinitely  long  rods  in  the  \\nc  AB  produced,  one  of  which 
is  attractive  and  the  other  repulsive,  are  the  series  of  ellipses 
described  from  the  extremities,  A  and  B,  as  foci,  while  the  surfaces 
of  equilibrium  are  generated  by  the  revolution  of  the  confocal 
hyperbolas- 


I8^  ABSTRACT  DYNAMICS. 

500.  As  of  immense  importance,  in  the  theory  not  only  of  gra- 
vitation but  of  electricity,  of  magnetism,  of  fluid  motion,  of  the 
conduction  of  heat,  etc.,  we  give  here  an  investigatiou  of  the  most 
important  properties  of  the  Potential. 

501.  This  function  was  introduced  for  gravitation  by  Laplace, 
but  the  name  \/as  first  given  to  it  by  Green,  who  may  almost  be 
said  to  have  created  the  theory,  as  w6  now  have  it.  Green's  work 
was  neglected  all  1846,  and  before  that  time  most  of  its  important 
theorems  had  been  re-discovered  by  Gauss,  Chasles,  Sturm,  and 
Thomson. 

In  §  245,  the  potential  energy  of  a  conservative  system  in  any  con« 
figuration  was  defined.  When  the  forces  concerned  are  forces  acting, 
either  really  or  apparently,  at  a  distance,'  as  attraction  of  gravitation, 
or  attractions  or  repulsions  of  electric  or  magnetic  origin,  it  is  in 
general  most  convenient  to  choose,  for  the  zero  configuration,  infinite 
distance  between  the  bodies  concerned.  We  have  thus  the  following 
definition : — • 

502.  The  mutual  potential  energy  of  t\vo  bodies  in  any  relative 
position  is  the  amount  of  work  obtainable  from  their  mutual  repulsion, 
hy  allowing  them  to  separate  to  an  infinite  distance  asunder.  When 
the  bodies  attract  mutually,  as  for  instance  when  no  other  force  than 
gravitation  is  operative,  their  mutual  potential  energy,  according  to 
the  convention  for  zero  now  adopted,  is  negative,  or  their  exhaustion 
cj potential  energy  is  positive. 

503.  The  Potential  at  any  point,  due  to  any  attracting  or  repelling 
body,  or  distribution  of  matter,  is  the  mutual  potential  energy  betv;een 
it  and  a  unit  of  matter  placed  at  that  point.  But  in  the  case  of 
gravitation,  to  avoid  defining  the  potential  as  a  negative  quantity, 
it  is  convenient  to  change  the  sign.  Thus  the  gravitation  potential, 
at  any  point,  due  to  any  mass,  is  the  quantity  of  work  reqyired  to 
remove  a  unit  of  matter  from  that  point  to  an  infinite  distance. 

504.  Hence,  if  V  be  the  potential  at  any  point  P,  and  V^  that  at 
a  proximate  point  Q,  it  evidently  follows  from  the  above  definition 
that  V—  V^  is  the  work  required  to  remove  an  independent  unit  of 
matter  from  P  Xo  Q;  and  it  is  useful  to  note  that  this  is  altogether 
independent  of  the  form  of  the  path  chosen  between  these  two  points, 
as  it  gives  us  a  preliminary  idea  of  the  power  v/e  acquire  by  the 
introduction  of  this  mode  of  representation. 

Suppose  Q  to  be  so  near  to  P  that  the  attractive  forces  exerted  on 
unit  of  matter  at  these  points,  and  therefore  at  any  point  in  the  line 
PQ,  may  be  assumed  to  be  equal  and  parallel.  Then  if  P  represent 
the  resolved  part  of  this  force  along  PQ,  P.  PQ  is  the  work  required 
to  transfer  unit  of  matter  from  P  to  Q.     Hence 

V-V,^P.PQ, 

V-  V 
F=\^ — li 
PQ 


STATICS   OF  A   PARTICLE.— ATTRACTION:      183 

that  is,  the  attraction  on  unit  of  matter  at  P  in  any  direction  PQ, 
is  the  rate  at  which  the  potential  at  P  increases  per  unit  of  lengtli 
oiPQ. 

505.  A  surface,  at  every  point  of  which  the  potential  has  the  same 
value,  and  therefore  called  an  Eqnipotential  Surface,  is  such  that  the 
attraction  is  everywhere  in  the  direction  of  its  normal.  For  in  no 
direction  along  the  surface  does  the  potential  change  in  value,  and 
therefore  there  is  no  force  in  any  such  direction.  Hence  if  the 
attracted  particle  be  placed  on  such  a  surface  (supposed  smooth  and 
rigid),  it  will  rest  in  any  position,  and  the  surface  is  therefore  some- 
times called  a  Surface  of  Equilibrium.  We  shall  see  later,  that  the 
force  on  a  particle  of  a  liquid  at  the  free  surface  is  always  in  the 
direction  of  the  normal,  hence  the  term  Lei'cl  Surface,  which  is  often 
used  for  the  other  terms  above. 

508.  If  a  series  of  equipotential  surfaces  be  constructed  for  values 
of  the^potential  increasing  by  equal  small  amounts,  it  is  evident  from^ 
§  504  that  the  attraction  at  any  point  is  inversely  proportional'  to 
the  normal  distance  between  two  successive  surfaces  close  to  that 
point;  since  the  numerator  of  the  expression  for  F  is,  in  this  case, 
constant. 

507.  A  line  drawn  from  any  origin,  so  that  at  every  point  of  its 
length  its  tangent  is  the  direction  of  the  attraction  at  that  point,  is 
called  a  Line  of  Force;  and  it  obviously  cuts  at  right  angles  every 
equipotential  surface  which  it  meets. 

These  three  last  sections  are  true  tvliatrcer  be  the  law  of  attraction ; 
in  the  next  we  are  restricted  to  the  law  of  the  inverse  square  of  the 
distance. 

508.  If,  through  every  point  of  the  boundary'  of  an  infinitely 
small  portion  of  an  equipotential  surface,  the  corresponding  lines  of 
force  be  drawn,  we  shall  evidently  have  a  tubular  surface  of  infinitely 
small  section.  The  resultant  force,  being  at  every  point  tangential 
to  the  direction  of  the  tube,  is  inversely  as  its  normal  transverse 
section. 

This  is  an  immediate  consequence  of  a  most  important  theorem, 
which  will  be  proved  later.  The  surface  integral  of  the  attraction 
exerted  by  any  distribution  of  matter  in  the  direction  of  the  riormal  at 
every  point  of  any  closed  surface  is  ^ttJI;  wJiere  J/  is  the  amount  of 
matter  within  the  surface,  while  the  attraction  is  considered  positive  or 
negative  according  as  it  is  inwards  or  outwards  at  any  point  of  the 
surface. 

For  in  the  present  case  the  force  perpendicular  to  the  tubular 
part  of  the  surface  vanishes,  and  we  need  consider  the  ends  only. 
When  none  of  the  attracting  imass  is  within  the  portion  of  the  tube 
considered,  we  have  at  once 

F'u:  -  F-M  =  o, 
/'being  the  force  at  any  point  of  the  section  whose  area  is  z7. 

This  is  equivalent  to  the  celebrated  equation  of  Laplace. 


i84  ABSTRACT  DYNAMICS. 

When  the  attracting  body  is  symmetrical  about  a  point,  the  lines 
of  force  are  obviously  straight  lines  drawn  from  this  point.  Hence 
the  tube  is  in  this  case  a  cone,  and,  by  §  486,  -m  is  proportional  to 
the  square  of  the  distance  from  the  vertex.  Hence  F  is  inversely 
as  the  square  of  the  distance  for  points  external  to  the  attracting 
mass. 

When  the  mass  is  symmetrically  disposed  about  an  axis  in  in- 
finitely long  cylindrical  shells,  the  lines  of  force  are  evidently  perpen- 
dicular to  the  aj:is.  Hence  the  tube  becomes  a  ivedge,  whose  section 
is  proportional  to  the  distance  from  the  axis,  and  the  attraction  is 
therefore  inversely  as  the  distance  from  the  axis. 

When  the  mass  is  arranged  in  infinite  parallel  planes,  •  each  of 
uniform  density,  the  lines  of  force  are  obviously  perpendicular  to 
these  planes;  the  tube  becomes  a  cylinder;  and,  since  its  section  is 
constant,  the  force  is  the  same  at  all  distances. 

If  an  infinitely  small  length  /  of  the  portion  of  the  tube  .considered 
pass  through  matter  of  density  p,  and  if  to  be  the  area  of  the  section, 
of  the  tube  in  this  part,  we  have 

Pu:  ~  F''uj  —  4:r/a)p. 
This  is  equivalent  to  Poisson's  extension  of  Laplace's  equation, 

509.  In  estimating  work  done>  against  a  force  which  varies  in- 
versely as  the  square  of  the  distance  from  a  fixed  point,  the  mean 
force  is  to  be  reckoned  as  the  geometrical  mean  between  the  forces 
at  the  beginning  and  end  of  the  path:  and,  whatever  may  be  the 
path  followed,  the  efiective  space  is  to  be  reckoned  as  the  difference 
of  distances  from  the  attracting  point.  Thus  the  work  done  in  any 
course  is  equal  to  the  product  of  the  difference  of  distances  of  the 
extremities  from  tlie  attracting  point,  into  the  geometrical  mean  of 
the  forces  at  these  distances;  or,  if  O  be  the  attracting  point,  and  in 
its  force  on  a  unit  mass  at  unit  distance,  the  work  don^  in  moving  a 
particle,  of  unit  mass,  from  any  position  P  to  any  other  position^ 
P\  is 

iOP-OP)  ^-^^r-^^,  or -^^^^,. 

To  prove  this  it  is  only  necessary  to  remark,'  that-  for  any  infinitely 
small  step  of  the  motion,  the  effective  space  is  clearly  the  difference 
of  distances  from  the  centre,  and  the  working  force  may  be  taken  as 
the  force  at  either  end,  or  of  any  intermediate  value,  the  geometrical 
mean  for  instance:  and  the  preceding  expression  applied  to  each 
infinitely  small  step  shows  that  the  same  rule  holds  for  the  sum 
making  up  the  whole  work  done  through  any  finite  range,  and  by 
any  path. 

Hence,  by  §  503,  it  is  obvious  that  the  potential  at  P,  of  a  mass  m 

.situated  at  O,  is  --- ;  and  thus  that  the  potential  of  any  mass  at  a 

point  P  is  to  be  found  by  adding  the  quotients  of  every  portion  of 
the  mass,  each  divided  by  its  distance  from  P. 


STATICS  OF  A  PARTICLE— ATTRACTION.      185 

510.  Let  S  be  any  closed  surface,  and  let  O  be  a  point,  either 
external  or  internal,  where  a  mass,  m,  of  matter  is  collected.  Let  N 
be  the  component  of  the  attraction  of  m  in  the  direction  of  the 
normal  drawn  inwards  from  any  point  P,  oi  S.  Then,  if  ^o-  denotes 
an  element  of  S,  and  //  integration  over  the  whole  of  it, 

//  Ndd  =  4r;//,  or  =  o, 

according  as  0  is  Internal  or  external. 

Case  /,  O  internal.  Let  OP^PP^..,  be  a  straight  line  draun  in 
any  direction  from  O,  cutting  S  in  jP, ,  /*,,  P^,  etc.,  and  therefore 
passing  out  at  P^,  in  at  P^,  out  again  at  P^,  in  again  at  P^,  and  so 
on.  Let  a  conjcal  surface  be  described  by  lines  through  O,  all  in- 
finitely near  OP^P^...,  and  let  w  be  its  solid  angle  (§  482).  The 
portions  of  ffNIcr  corresponding  to  the  elements  cut  from  ^  by 
this  cone  will  be  clearly  each  equal  in  absolute  magnitude  to  ww, 
but  will  be  alternately  positive  and  negative.  Hence  as  there  is  an 
odd  number  of  them,  their  sum  is  +  am.  And  the  sum  of  these,  for 
all  solid  angles  round  (9,  is  (§483)  equal  to  47r;«;  that  is  to  say, 

Case  11.^  O  external.  Let  OP^P^Py..  be  a  line  drawn  from  O  pass* 
ing  across  S,  inwards  at  P,,  outwards  at  P.,,  and  so  on.  Drawing, 
as  before,  a  conical  surface  of  infinitely  small  solid  angle,  to,  we  have 
still  WW  for  the  absolute  value  of  each  of  the  portions  of  fjJVdcr 
corresponding  to  the' elements  which  it  cuts  from  S;  but  their  signs 
are  alternately  negative  and  positive ;  and  therefore  as  their  number 
is  even,  their  sum  is  zero.     Hence 

JJJVd<T^O. 

Tvorh  these  results  it  follows  immediately  that  if  there  be  any  con- 
tinuous distribution  of  matter,  partly  within  and  partly  without  a 
closed  surface  S,  and  IV  ^nd  da-  be  still  used  with  the  same  signifi- 
cation, we  have 

//iWo-  =  4TrM 

if  M  denote  the  whole  amount  of  matter  within  ^ 

511.  From  this  it  follows  that  the  potential  cannot  have  a  maxi- 
mum or  minimum  value  at  a  point  in  free  space.  For  if  it  were  so, 
a  closed  surface  could  be  described  about  the  point,  and  indefinitely 
near  it,  so  that  at  every  point  of  it  the  value  of  the  potential  would  be 
less  than,  or  greater  than,  that  at  the  point;  so  that  JV  would  be 
negative  or  positive  all  over  the  surface,  and  therefore  JJJVIcr  would 
be  finite,  which  is  impossible,  as  the  surface  contains  none  of  the 
attracting  mass. 

512.  It  is  also  evident  that  iV must  have  positive  values  at  some 
parts  of  this  surface,  and  negative  values  at  others,  unless  it  is  zero 
all  over  it.  Hence  in  free  space  the  potential,  if  not  constant  round 
any  point,  increases  in  some  directions  from  it,  and  diminishes  in 


1 86  ABSTRACT  DYNAMICS. 

others;  and  therefore  a  material  particle  placed  at  a  point  of 
zero  force  under  the  action  of  any  attracting  bodies,  and  free 
from  all  constraint,  is  in  unstable  equilibrium,  a  result  diie  to 
Earnshaw^ 

513.  If  the  potential  be  constant  over  a  closed  surface  which 
contains  none  of  the  attracting- mass,  it  has  the  same  constant  value 
throughout  the  interior.  For  if  not,  it  must  have. a  maximum  or 
minimum  value  somewhere  within,  which  is  impossible. 

514.  The  mean  potential  over' any  spherical  surface,  due  to  matter 
entirely  without  it,  is  equal  to  the  potential  at  its  centre;  a  theorem 
apparently  first  given  by  Gauss.  See  also  Cambridge  Mathematical! 
Journal,  Feb.  1845  (vol.  iv.  p.  225).  This  proposition  is  merely  an 
extension,  to  any  masses,  of  the  converse  of  the  following  statement, 
which  is  easily  seen  to  follow  from  the  results  of  §§  479,  488  expressed 
in  potentials  instead  of  forces.  The'potential  of  an  uniform  spherical 
shell  at  an  external,  point  is  the  same  as  if  its  mass  were  condensed 
at  the  centre.  At  all  internal  points  it  has  the  same  value  as  at  the 
surface. 

515.  If  the  potential  of  any  masses  has  a  constant  value,  V, 
through  any  finite  portion,  ^,  of  space,  unoccupied  by  matter,  it  is 
equal  to  F  through  every  part  of  space  which  can  be  reached  in  any 
way  without  passing  through  any  of  those  masses  :  a  very  remarkable 
proposition,  due  to  Gauss.  For,  if  the  potential  differ  from  V  in  • 
space  contiguous  to  K,  it  must  (§  513)  be  greater  in  some  parts  and 
less  in  others. 

From  any  point  C- within  K,  as  centre,  in  the  neighbourhood  of  a 
place  where  the  potential  is  greater  than  V,  describe  a  spherical 
surface  not  large  enough  to  contain  any  part  of  any  of  the  attracdng- 
masses,  nor  to  include  any  of  the  space  external  to  K  except  such 
as  has  potential  greater  than  V.  But  this  is  impossible,  since  we 
have  just  seen  (§  514)  that  the  mean  potential  over  the  spherical 
surface  must  be  V-  Hence  the  supposition  that  the  potential  is 
greater  than  V  in  some  places  and  less  in  others,  contiguous  to  K 
and  not  including  masses,  is  false. 

516.  Similarly  we  see  that  in  any  case  of  symmetry  round  an 
axis,  if  the  potenrial  is.  constant  through  a  certain  finite  distance, 
however  short,  along  the  axis,  it  is  constant  throughout  the  whole 
space  that-  can  be  reached  from  this  portion  of  the  axis,  without 
crossing  any  of  the  masses. 

517.  Let  .S  be  any  finite  portion  of  a  surface,  or  Complete  closed 
surface,  or  infinite  surface,  and  let  E  be  any  point  on  S.  (a)  It  is 
possible  to  distribute  matter  over  S  so  as  to  produce  potential  equal 
to  E  {E),  any  arbitrary  function  of  the  position  of  E,  over  the  whole 
of  S.  {b)  There  is  only  one  whole  quantity  of  matter,  and  one 
distribution  of  it,  which  can  satisfy  this  condition.  •  For  the  proof  of 

*  Cambridge  Phil.  Trans.,  March,  1839. 


STATICS  OF  A  PARTICLE.— ATTR ACTIO X.      1S7 


this  and  of  several  succeeding  theorems,  \ve  refer  the  reader  to  our 
larger  work. 

518.  It  is  important  to  remark  that,  if  6"  consist,  in  part,  of  a 
closed  surface,  <2,  the  determination  of  c7,  the  potential  at  any  point, 
Avithki  it  -will  be  independent  of  those  portions  of  S,  if  any,  wliich 
lie  without  it;  and,  c'/l-j  versa,  the  determination  of  6^"  through  external 
space  will  be  independent  of  those  portions  of  S,  if  any,  which  lie 
v.ithin  the  part  Q.  Or  if  S  consist,  in  part,  of  a  surface  Q,  extend- 
ing infinitely  in  all  directions,  the  determination  of  U  through  all 

■  space  on  either  side  of  Q,  is  independent  of  those  portions  oi  S,  if 
any,  which  lie  on  the  other  side.  ■ 

519.  Another  remark  of  extreme  imiportance  is  this: — If  F {E) 
be  the  potential  at  E  of  any  distribution,  y>/,  of  matter,  and  if  S  be 
such  as  to  separate  perfectly  any  portion  or  portions  of  space,  H^ 
from  all  of  this  matter;  that  is  to  say,  such  that  it  is  impossible  to 
pass  into  //from  any  part  of  J/ without  crossing  S;  then,  throughout 
H,  the_  value  of  6"  will  be  the  potential  oi  M. 

520.  Thus,  for  instance,  if  6"  consist  of  three  detached  surfaces, 
^j,  6'^,  S^,  as  in' the  diagram,  of  which  S^,  S.^  are  closed,  and  S^  is 
an  open  shell,  and  if  F  (E)  be 
the  potential  due  to  M,  at  any 
point,  E,  of  any  of  these  portions 
of  S;  then  throughout  ZTj  and 
Hj,  the  spaces  within  S^  and  with- 
out S.-^,  the  value  of  U"  is  simply 
the  potential  of  M.  The  yalue  of 
U"  tlirough  A',  the  remainder  of 
space,  depends,  of  course,  on  the 
character  of  the  composite  sur- 
face S. 

521.  From  §  518  follov.-s  the  grand  proposition  : — //  is  possible  io 
find  one,  hut  no  ether  than  cne,  distribution  of  matter  over  a  surface  S 

which  shall  produce  over  S,  and  throughout  all  space  H  separated  by  S 
from  every  part  of  M,  the  same  potential  as  any  given  mass  M. 

Thus,  in  the  preceding  diagram,  it  is  possible  to  find  one,  and  but 
one,  distribution  of  matter  over  ^j,  S^,  S^  which  shall  produce  over 
^■3  and  through  H^  and  H^  the  same  potential  as  M. 

The  statement  of  this  proposition  most  commonly  made  is :  //  is 
possible  to  distribute  inatter  over  any  surface,  S,  completely  enclosing  a 
mass  M,  so  as  to  produce  the  same  potaiiial  as  M  through  all  space 
outside  M ;  which,  though  seemingly  more  limited,  is,  when  inter- 
preted with  proper  mathematical  comprehensiveness,  equivalent  to  the 
foregoing. 

522.  If  S  consist  of  several  closed  or  infinite  surfaces,  .S^,  S^^  S^y 
respectively  separating  certain  isolated  spaces  ZTj,  //g,  /f,,  from  Ify 
the  remainder  of  all  space,  and  if  F  {E)  be  the  potential  of  masses 
Wi,  w^,  Wg,  lying  in  the  spaces  /fj,  H^y  JI./,  the  portions  of  6/ due  to 


ABSTRACT  DYNAMICS. 


Si,  S^,  S^,  respectively  will  throughout  If  be  equal  respectively  tO' 
the  potentials  of  Wi,  in^,m^,  separately. 


V    m 


Por,  as  we  have  just  seen,  it  is  possible  to  find  one,  but  only  one, 
distribution  of  matter  over  S,  which  shall  produce  the  potential  of  Wj, 

throughout  all  the  space  H,  H^, 
H^,  etc.,  and  one,  but  only  one, 
distribution  over  S^  which  shall 
produce  the  potential  of  Wj 
throughout  Z^  Hi,  H^,  etc. ;  and 
so  on.  But  these  distributions 
on  Si,  »Sg,  etc.,  jointly  constitute 
a  distribution  producing  the  po- 
tential F{E)  over  every  part  of 
S,  and  therefore  the  sum  of  the 
potentials  due  to  theni  all,  at  any  point,  fulfils  the  conditions  pre- 
sented for  U.    This  is  therefore  (§518)  the  solution  of  the  problem. 

523.  Considering  still  the  case  in  which  i^(^)  is  prescribed  to  be 
the  potential  of  a  given  mass,  J/:  let  S  be  an  equipotential  surface 
enclosing  M,  or  a  group  of  isolated  surfaces  enclosing  all  the  parts  of 
M,  and  each  equipotential  for  the  whole  of  M.  The  potential  due  to 
the  supposed  distribution  over  S  will  be  the  same  as  that  of  M, 
through  all  external  space,  and  will  be  constant  (§  514)  through  each 
enclosed  portion  of  space.  Its  resultant  attraction  will  therefore  be 
the  same  as  that  of  M  on  all  external  points,  and  zero  on  all  internal 
points.     Hence  we  see  at  once  that  the  density  of  the  matter  dis- 

tributed  over  it,  to  produce  F  {£},  is  equal  to  —  where  R  denotes 

the  resultant  force  of  M,  at  the  point  F, 

524.  When  il/' consists  of  two  portions  Wj  and  tn'  separated  by  an 
equipotential  Si,  and  S  consists  of  two  portions.  Si  and  S',  of  which 
the  latter  separates  the  former  perfectly  from  m' ;  we  see,  by  §  522, 
that  the  distribution  over  ,5,  produces  through  all  space  on  the  side 
of  it  on  which  S"  lies,  the  same  potential,  ^,  as  f«,,  and  the  dis- 
tribution on  S"  produces  through  space  on  the  side  of  it  on  which  Si 
lies,  the  same  potential,  F',  as  m'.  But  the  supposed  distribution  on 
the  wholeof  6"  is  such  as  to  produce  a  constant  potential,  Cy  over  .S,, 


STATICS   OF  A  PARTICLE.— ATTRACTION.     189. 

and  consequently  the  same  at  every  point  within  S^.  Hence  the  in- 
ternal potential,  due  to  S^  alone,  is  C^-V. 

Thus,  passing  from  potentials  to  attractions,  we  see  that  the  re- 
sultant attraction  of  S^  alone,  on  all  points  on  one  side  of  it,  is  the 
same  as  that  of  Wj ;  and  on  the  other  side  is  equal  and  opposite  to< 
that  of  the  remainder  m'  of  the  whole  mass.  The  most  direct  and 
simple  complete  statement  of  this  result  is  as  follows  : — 

If  masses  m,  7n',  in  portions  of  space,  H,  H\  completely  separated 
from  one  another  by  one  continuous  surface  S,  whether  closed  or 
infinite,  are  knonTi  to  produce  tangential  forces  equal  and  in  the 
same  direction  at  eacli  point  of  S,  one  and  the  same  distribution 
of  matter  over  S  will  produce  the  force  of  m  throughout  H',  and 
that  of  ot'  throughout  H.     The  density  of  this  distribution  is  equal  to 

— ,  if  i?  denote  the  resultant  force  due  to, one  of  the  masses,  and 
47r 

the  other  with  its  sign  changed.  And  it  is  to  be  remarked  that  the 
direction  of  this  resultant  force  is,  at  every  point,  i?,  of  S,  perpen- 
dicular to  S,  since  the  potential  due  to  one  mass,  and  the  other  with 
its  sign  changed,  is  constant  over  the  whole  of  S. 

526.  Green,  in  first  publishing  his  discovery  of  the  result  stated 
in  §  523,  remarked  tliat  it  shows  a  way  to  find  an  infinite  variety  of 
closed  surfaces  for  any  one  of  which  we  can  solve  the  problem  of 
determining  the  distribution  of  matter  over  it  which  shall  produce 
a  given  uniform  potential  at  each  point  of  its  surface,  and  con- 
sequently the  same  also  throughout  its  interior.  Thus,  an  example 
which  Green  himself  gives,  let  M  be  a  uniform  bar  of  matter,  AA'. 
The  equipotential  surfaces  round  it  are,  as  we  have  seen  above 
(§  499  (f)),  prolate  ellipsoids  of  revolution,  each  having  A  and  A'  for 
its  foci;  and  the  resultant  force  at  Cwas  found  to  be 

■.CF, 


l{l'-a') 

the  whole  mass  of  the  bar  being  denoted  by  w,  its  length  by  2a,  and 
A'C+  AChy  2I.  We  conclude  that  a  distribution  of  matter  over 
the  surface  of  the  ellipsoid,  having 

I     m.CF 

47r  l{l^-a-) 

for  density  at  C,  produces  on  all  external  space  the  same  resultant 
force  as  the  bar,  and  zero  force  or  a  constant  potential  through  the 
internal  space.  This  is  a  particular  case  of  the  general  result  re- 
garding ellipsoidal  shells,  proved  below,  in  §§  536,  537. 

526.     As  a  second  example,  let  J/ consist  of  two  equal  particles, 
at  points  /,  /'.     If  we  take  the  mass  of  each  as  unity,  the  potential  at 

/*  is  jp  +  -jrp;  and  therefore 

L^^1-=C 
IP    IP 


1^0  ABSTRACT  DYNAMICS. 

is  the  equation  of  an  equipotential  surface ;  it  being  understood  that 
negative  values  of  IF  and  I'P  are  inadmissible,  and  that  any  con- 
§t.^jit  value,  from  oo  to  o,  may  be  given  to  C.  The  curves  in  the 
annexed  diagram  have  been  drawn,  from  this  equation,  for  the  cases 
of  C  equal  respectively  to  lo,  9,  8,  7,  6,  5,  4-5,  4-3,  4-2,  4-1,  4,  3-9, 
3-8,  37,  3-5,  3,  2-5,  2  ;  the  value  of  //'  being  unity. 

The  corresponding  equipotential  surfaces  are  the  surfaces  traced 
by  these  curves,  if  the  whole  diagram  is  made  to  rotate  round  //'  as 


axis.    Thus  we  see  that  for  any  values  of  C  less  than  4  the  equi- 
potential surface  is  one  closed  surface.    Choosing  any  one  of  these 

surfaces,  let  7?  denote  the  resultant  of  forces  equal  to  jrp-^  and  -p-pt 

in  the  lines  PI  and  FT.    Then  if  matter  be  distributed  over  this 

surface,  with  density  at  F  equal  to  — ,  its  attraction  on  any  internal 

4"" 
point  will  be  zero;  and  on  any  external  point,  will  be  the  same  as 
that  of  /  and  /'. 

527.  For  each  value  of  C  greater  than  4,  the  equipotential  surface 
consj,sts  of  two  detached  ovals  approximating  (the  last  three  or  four 
in  the  diagram,  very  closely)  to  spherical  surfaces,  with  centres  lying 
between  the  points  /  and  /',  but  approximating  more  and  more 
closely  to  these  points,  for  larger  and  larger  values  of  C. 

Considering  one  of  these  ovals  alone,  one  of  the  series  enclosing 
/',  for  instance,  and  distributing  matter  over  it  according  to  the  same 

law  of  density,  — ,  we  have  a  shell  of  matter  which  exerts  (§  525) 


STATICS  OF  A  PARTICLE.— ATTRACTION.      191 


on  external  points  the  same  force  as  /';  and  on  internal  points  a 
force  equ.il  and  opposite  to  that  of  /. 

528.    As  an  example  of  exceedingly  great  importance  in  the  theory 
of  electricity,    let   M  consist   of  a  positi\-e  mass,  w,  concentrated 
at  a  point   /,  and   a    negative 
mass,  -  w',  at  /' ;   and   let   »S 
be  a  spherical   surface   cutting 
//'  and  //'  produced  in  points 
Ay  A^,  such  that 
lA  :AI'::IA,:I'A/.:m  :  m'. 

Then,  by  a  well-known  geo- 
metrical proposition,  we  shall 
have  IE  :  IE  ::  m  :  m' ',  and 
therefore 


IE     FE  * 

Hence,  by  what  we  have  just  seen,  one  and  the  same  distribution  of 
matter  over  S  will  produce  the  same  force  as  w'  througli  all  external 
space,   and  the   same  as  m  through  all  the  space  within  6".     And, 

finding  the  resultant  of  the  forces  ^^  in  Z"/,  and  -p-pi  in  FE,  pro- 
duced, which,  as  these  forces  are  inversely  as  IE  to  FE,  is  (§  222) 
■equal  to 

m         J.  J,        m^IF     I 

i^te    '  ^^  iir~  w  * 

we  conclude  that  the  density  in  the  shell  at  E  is 

m\IF      I 

47rw'  *  lE^  ' 
That  the  shell  thus  constituted  does  attract  external  points  as  if  its 
mass  were  collected  at  /',  and  internal  points  as  a  certain  mass  col- 
lected at  /,  was  proved  geometrically  in  §  491  above. 

529,     If  the  spherical  surface  is  given,  and  one  of  the  points,  /,  /', 

CA' 
-for  instance  /,  the  other  is  found  by  taking  CF  =  -^  >  ^'^'^  for  the 

mass  to  be  placed  at  it  we  have 

FA         CA         CF 
AI  CI  CA 

Hence,  if  we  have  any  number  of  particles  nt^,  m^,  etc.,  at  points 
/,,  /,,  etc.,  situated  without  S,  we  may  find  in  the  same  way  cor- 
responding internal  points  /'„  /',,  etc.,  and  masses  m\,in\,  etc.; 
and,  by  adding  the  expressions  for  the  density  at  E  given  for  each 
pair  by  the  preceding  formula,  we  get  a  spherical  shell  of  nvatter 
which  has  the  property  of  acting  on  all  external  space  with  the  same 
force  as  -;«',,  -ni\,  etc.,  and  on  all  internal  points  with  a  force 
•equal  and  opposite  to  that  of  Wj,  m^,  etc. 


193  ABSTRACT  DYNAMICS. 

530.  An  infinite  number  of  such  particles  may  be  given,  con- 
stituting a  continuous  mass  M',  when  of  course  the  corresponding 
internal  particles  will  constitute  a  continuous  mass,  —  M\  of  the 
opposite  kind  of  matter;  and  the  same  conclusion  will  hold.  If  5  is 
the  surface  of  a  solid  or  hollow  metal  ball  connected  with  the  earth 
by  a  fine  wire,  and  yl/an  external  influencing  body,  the  shell  of  matter 
we  have  determined  is  precisely  the  distribution  of  electricity  on  S 
called  out  by  the  influence  of  M:  and  the  mass  -  M\  determined  as 
above,  is  called  the  Electric  Image  of  M  in  the  ball,  since  the  electric 
action  through  the  whole  space  external  to  the  ball  would  be 
unchanged  if  the  ball  weie  removed  and  —  M'  properly  placed  \n 
the  space  left  vacant.  We  intend  to  return  to  this  subject  under 
Electricity. 

531.  ,  Irrespectively  of  the  special  electric  application,  this  method 
of  images  gives  a  remarkable  kind  of  transformation  which  is  often 
useful,  It  suggests  for  mere  geometry  what  has  been  called  the 
transfciTnation  by  reciprocal  radius-vectors ;  that  is  to  say,  the  sub- 
stitution for  any  set  of  points,  or  for  any  diagram  of  lines  or  surfaces, 
another  obtained  by  drav/ing  radii  to  them  from  a  certain  fixed  point 
or  origin,  and  measuring  off  lengths  inversely  proportional  to  these 
radii  along  their  directions.  We  see  in  a  moment  by  elementary 
geometry  that  any  line  thus  obtained  cuts  the  radius-vector  through 
any  point  of  it  at  the  same, angle  and  in  the  same  plane  as  the  line 
from  which  it  is  derived.  Hence  any  two  lines  or  surfaces  that  cut 
one  another  give  two  transformed  lines  or  surfaces  cutting  at  the 
same  angle :  and  infinitely  small  lengths,  areas,  and  volumes  trans- 
form into  others  whose  magnitudes  are  altered  respectively  in  '"the 
ratios  of  the  first,  second,  and  third  powers  of  the  distances  of  the 
latter  from  the  origin,  to  the  same  powers  of  the  distances  of  the 
former  from  the  same.  Hence  the  lengths,  areas,  and  volumes  in 
the  transformed  e  diagram,  corresponding  to  a  set  of  given  equal 
infinitely  small  lengths,  areas,  and  volumes,  however  situated,  at 
diff'erent  distances  from  the  origin,  are  inversely  as  the  squares,  the 
fourth  powers  and  the  sixth  powers  of  these  distances.  Further,  it  is 
easily  proved  that  a  straight  line  and  a  plane  transform  into  a  circle 
and  a  spherical  surface,  each  passing  through  the  origin ;  and  that, 
generally,  circles  and  spheres  transform  into  circles  and  spheres. 

532.  In  the  theory  of  attraction,  the  transformation  of  masses, 
densities,  and  potentials  has  also  to  be  considered,  Thus,  according 
to  the  foundation  of  the  method  (§  530),  equal  masses,  of  infinitely 
small  dimensions  at  different  distances  from  the  origin,  transform  into 
masses  inversely  as  these  distances,  or  directly  as  the  transformed 
distances  :  and,  therefcye,  equal  densities  of  lines,  of  surfaces,  and  of 
solids,  given  at  any  stated  distances  from  the  origin,  transform  into 
densities  directly  as  the  first,  the  third,  and  the  fifth  powers  of  those 
distances ;  or  inversely  as  the  same  powers  of  the  distances,  from 
the  origin,  of  the  corresponding  points  in  the  transformed  system. 
The  usefulness  of  this  transformation  in  the  theory  of  electricity, 


STATICS   OF  A   PARTICLE.— ATTRACTION". 


193 


and  of   attraction  in  general,  depends  entirely  on  the    following 
theorem  : — 

Let  ^  denote  the  potential  at  P  due  to  the  given  distribution,  and  (^' 
the  potential  at  F  due  to  the  transformed  distribution :  then  shall 

<ft'  —  -  (f>  =  -,  <f}. 

a        r 

Let  a  mass  w  collected  at  /  be  any  part  of  the  given   distri- 
bution, and   let  7rt    at  /'  be 
the     corresponding     part    in  P 

the  transformed  distribution.  j     ^ 

We  have 


a'  =  OT.OI=OP'.OP, 
and  therefore 

01 :  OP ::  OP' '.  or ;        0  I'  I 

which  shows  that  the  triangles  IPO,  P'TO  are  similar,  so  that 

IP  :  PT  ::  JOIOP:  JOPTOT  ::  OIOP  :  a\ 
We  have  besides 

VI  ;  7/1" : :  01 :  a, 
and  therefore 

in       in 

Jp--Tpr---a'.OP 

Hence  each  term  of  ^  bears  to  the  corresponding  term  of  ^' 
the  same  ratio;  and  therefore  the  sum,  </>,  must  be  to  the  sum, 
^',  in  that  ratio,  as  was  to  be  proved. 

533.  As  an  example,  let  the  given  distribution  be  confined  to  a 
spherical  surface,  and  let  O  be  its  centre  and  a  its  own  radius.  The 
transformed  distribution  is  the  same.  But  the  space  within  it  becomes 
transformed  into  the  space  without  it.  Hence  if  <^  be  the  potential 
due  to  any  spherical  shell  at  a  point  P,  within  it,  the  potential  due 

to  the  same  shell  at  the  point  P"  in  OP  produced  till  OP'  =  -^p,  is. 

equal  to  -^p,  ^  (which  is  an  elementary  proposition  in  the  spherical 

harmonic  treatment  of  potentials,  as  we  shall  see  presently).  Thus, 
for  instance,  let  the  distribution  be  uniform.  Then,  as  we  know 
there  is  no  force  on  an  interior  point,  <l>  must  be  constant;  and 
therefore  the  potential  at  P',  any  external  point,  is  inversely  propor- 
tional to  its  distance  from  the  centre. 

Or  let  the  given  distribution  be  a  uniform  shell,  ,5*,  and  let  0  be 
any  eccentric  or  any  external  point.  The  transformed  distribution 
becomes  (§^  531,  532)  a  spherical  shell,  S',  with  density  varying 
inversely  as  the  cube  of  the  distance  from  O.  If  O  is  within  S,  it  is 
also  enclosed  by  S'^  and  the  whole  space  within  S  transforms  into 


194  ABSTRACT  DYNAMICS. 

the  whole  space  without  S'.  Hence  (^  532)  the  i^otential  of  S'  at 
any  point  without  it  is  inversely  as  the  distance  from  O,  and  is  there- 
fore that  of  a  certain  quantity  of  matter  collected  at  O.  Or  if  O  is 
external  to  6",  and  consequently  also  external  to  S',  the  space  within 
S  transforms  into  the  space  within  S'.  Hence  the  potential  of  S' 
at  any  point  within  it  is  the  same  as  that  of  a  certain  quantity  of 
matter  collected  at  O,  which  is  now  a  point  external  to  it.  Thus, 
without  taking  advantage  of  the  general  theorems  (§§  517,  524),  we 
fall  back  on  the  same  results  as  m'c  inferred  from,  them  in  §  528,  and 
as  we  proved  synthetically  earlier  (§§  488,  491,  492),  It  may  be 
remarked  that  those  synthetical  demonstrations  consist  merely  of 
transformations  of  Newton's  demonstration,  that  attractions  balance 
on  a  point  within  a  uniform  shell.  Thus  the  first  of  them  (§  4S8) 
is  the  image  of  Newton's  in  a  concentric  spherical  surface;  and  the 
second  is  its  image  in  a  spherical  surface  having  its  centre  external 
to  the  shell,  or  internal  but  eccentric,  according  as  the  first  or  the 
second  diagram  is  used. 

531.  We  shall  give  just  one  other  application  of  the  theorem 
of  §  532  at  present,  but  much  use  of  it  will  be  made  later  in  the 
theory  of  Electricity. 

Let  the  given  distribution  of  mattjer  be  a  uniform  solid  sphere,  B, 
and  let  O  be  external  to  it.  The  transformed  system  will  be  a  solid 
sphere,  B' ,  with  density  varying  inversely  as  the  fifth  power  of  the 
distance  from  O,  a  point  external  to  it.  The  potential  of  B  is  the 
same  throughout  external  space  as  that  due  to  its  mass,  in,  collected 
at  its  centre,  C.  Hence  the  potential  of  B'  through  space  external 
to  it  is  tlie  same  as  that  of  the  corresponding  quantity  of  matter 
collected  at  C,  the  transformed  position  of  C.  This  quantity  is  of 
course  equal  to  the  mass  of  B'.  And  it  is  easily  proved  that  C  is 
the  position  of  the  image  of  O  in  the  spherical  surface  of  B'.  We 
conclude  that  a  soHd  sphere  with  density  varying  inversely  as  the 
fifth  power  of  the  distance  from  an  external  point,  O,  attracts  any 
external  point  as  if  its  mass  were  condensed  at  the  image  of  O  in  its 
external  surface.  It  is  easy  to  verify  this  for  points  of  the  axis  by 
direct  integration,  and  thence  the  general  conclusion  follows  ac- 
cording to  §  508. 

535.  The  determination  of  the  attraction  of  an  ellipsoid,  or  of  an 
ellipsoidal  shell,  is  a  problem  of  great  interest,  and  its  results  will  be 
of  great  use  to  us  afterwards,  especially  in  Magnetism.  \\^e  have 
left  it  till  now,  in  order  that-  we  may  be  prepared  to  apply  the  pro- 
perties of  the  potential,  as  they  afford  an  extremely  elegant  method 
of  treatment.     A  few  definitions  and  lemmas  are  necessary. 

Corresponding poinis  on  two  confocal  ellipsoids  arc  such  as  coincide 
when  either  ellipsoid  by  a  pure  strain  is  defornaed  so  as  to  cpincide 
with  the  other. 

And  it  is  easily  shown,  that  if  any  two  points,  P,  Q,  be  assumed  on 
one  shell,  and  their  correspomU^ig  points,  A  q,  on  the  other,  we  have 
'Pq^Qp. 


STATICS   OF  A   PARTICLE—ATTRACTION,     195 

The  species  of  shell  which  it  is  most  convenient  to  employ  in  the 
subdivision  of  a  homogeneous  ellipsoid  is  bounded  by  similar,  simi- 
larly situated,  and  concentric  ellipsoidal  surfaces;  and  it  is  evident 
from  the  properties  of  pure  strain  (§  141)  that  such  a  shell  may  be 
produced  from  a  spherical  shell  of  uniform  thickness  by  uniform 
extensions  and  compressions  in  three  rectangular  directions.  Unless 
the  contrary  be  specified,  the  word  'shell'  in  connexion  with  this 
■subject  will  always  signify  an  infinitely  thin  shell  of  the  kind  now 
described. 

536.  Since,  by  §  479,  a  homogeneous  spherical  shell  exerts  no 
attraction  on  an  internal  point,  a  homogeneous  shell  (which  need 
not  be  infinitely  thin)  bounded  by  similar,  and  similarly  situated,  and 
concentric  ellipsoids,  exerts  no  attraction  on  an  internal  point, 

For  suppose  the  spherical  shell  of  §  479,  by  simple  extensions  and 
compressions  in  three  rectangular  directions,  to  be  transformed  into 
an  ellipsoidal  shell.  In  this  distorted  form  the  masses  of  all  parts 
are  reduced  or  increased  in  the  proportion  of  the  mass  of  the  ellipsoid 
to  that  of  the  sphere.  Also  the  ratio  of  the  lines  HP,  PK  is  un- 
altered, §  139.  Hence  the  elements  IH^  KL  still  attract  /'equally, 
and  the  proposition  follows  as  in  §  479. 

Hence  inside  the  shell  the  potential  is  constant 

537.  Two  confocal  shells  (§  535)  being  given,  the  potential  of  the 
first  at  any  point,  P^  of  the  surface  of  the  second,  is  to  that  of  the 
second  at  the  corresponding  point,/,  on  the  surface  of  the  first,  as 
the  mass  of  the  first  is  to  the  mass  of  the  second.  This  beautiful 
proposition  is  due  to  Chasles. 

To  any  element  of  the  mass  of  the  outer  shell  at  Q  corresponds  an 
•element  of  mass  of  the  inner  at  q,  and  these  bear  the  same  ratio  to 
the  whole  masses  of  their,  respective  shells,  that  the  corresponding 
element  of  the  spherical  shell  from  which  either  may  be  derived  bears 
to  its  whole  mass.  Whence,  since  Pq  =  Qp,  the  proposition  is  true 
for  the  corresponding  elements  at  Q  and  ^,  and  therefore  for  the 
entire  shells. 

Also,  as  the  potential  of  a  shell  on  an  internal  point  is  constant, 
and  as  one  of  two  confocal  ellipsoids  is  wholly  within  the  other :  it 
follows  that  the  external  equipotential  surfaces  for  any  such  shell  are 
confocal  ellipsoids,  and  therefore  that  the'attraction  of  the  shell  on  an 
external  point  is  normal  to  a  confocal  ellipsoid  passing  through  the 
point. 

538.  Now  it  has  been  shown  (§  495)  that  the  attraction  of  a  shell 
on  an  external  point  near  its  surface  exceeds  that  on  an  internal 
point  infinitely  near  it  by  477^  where  p  is  the  surface-density  of  th.e 
shell  at  that  point.  Hence,  as  (^  536)  there  is  no  attraction  on  an 
internal  point,  the  attraction  of  a  shell  on  a  point  at   iis  exterior 

*;surface  is  4rrp:  or  4rp/  if  p  be  now  put  for  the  voluinc-dcnsity,  and  / 
for  the  (infinitely  small)  thickness  of  th$  shell,  §  495.  »Froni  tliis 
it  is  easy  to-  obtain  by  integration  the  determination  of  the  whole 
attraction  of  a  homogeneous  ellipsoid  on  an  externa!  particle.* 


igS  ABSTRACT  DYNAMICS. 

530.    The  following  splendid  theorem  is  due  to  Maclaurin : — 

21ie  aitradlons  exerted  by  two  homogeneous  and  confocal  ellipsoids 
on  the  same  point  external  to  each,  or  external  to  one  and  on  the  sicr- 
face  of  the  other ^  are  in  the  same  direction  and  proportional  to  their 
masses, 

540.  Ivory's  theorem  is  as  follows  :— 

Let  correspofidmg  points  P,  /,  be  taken  on  the  surfaces  of  ttvo  homo- 
geneous  confocal  ellipsoids,  E,  e.  The  x  componetit  of  the  attraction  of  E 
on  p,  is  to  that  of  e  on  P  as  the  area  of  the  section  of  E  by  the  plane  of 
yz  is  to  that  of  th6  co}lanar  section  of  e. 

Poisson  showed  that  this  theorem  is^  true  for  any  law  of  force 
whatever.  This  is  easily  proved  by  employing  in  the  general  ex- 
pressions for  the  components  of  the  attraction  of  any  body,  after  one 
integration,  the  properties  of  corresponding  points  upon  confocal 
ellipsoids  (§  535). 

541.  An  ingenious  application  of  Ivory's  theorem,  by  Duhamel, 
must  not  be  omitted  here.  Concentric  spheres  are  a  particular  case 
of 'confocal  ellipsoids,  and  therefore  the  attraction  of  any  sphere  on 
a  point  on  the  surface  of  ap  internal  concentric  sphere,  is  to  that  of 
the  latter  upon  a  point  in  the  surface  of  the  former  as  the  squares  of 
the  radii  of  the  spheres.  Now  if  the  law  of  attractioit  be  such  that  a 
homogeneous  spherical  shell  of  uniform  thickness  exerts  ?io  attraction  on 
an  internal  point,  the  action  of  the  larger  sphere  on  the  internal  point 
is  reducible  to  that  of  the  smaller.  Hence  the  law  is  that  of  the  in' 
verse  square  of  the  distance,  as  is  easily  seen  by  making  the  smaller 
sphere  less  and  less  till,  it  becom.es  a  mere  particle.  This  theorem  is 
due  originally  to  Cavendish. 

542.  {Definition.)  If  the  action  of  terrestrial  or 'Other  gravity  on 
a  rigid  body  is  reducible  to  a  single  force  in  a  line  passing  always 
through  one  point  fixed  relatively  to  the  body,  whatever  be  its  position 
relatively  to  the  earth-  or  other  attracting  mass,  that  point  is  called  its- 
ce?itre  of  gravity,  and  the  body  is  called  a  ccntrobaric  body. 

543.  One  of  the  most  startling  results  of  Green's  wonderful  theory 
of  tlie  potential  is  its  establishment  of  the  existence  of  centrobaric 
bodies;  and  the  discovery  of  their  properties  is  not  the  least  curious 
and  interesting  among  its  very  various  applications. 

544.  If  a  body  {B)  is  centrobaric  relatively  to  any  one  attracting 
mass  {A),  it  is  centrobaric  relatively  to  every  other:  and  it  attracts 
all  matter  external  lo  itself  as  if  its  own  mass  were  collected  in  its 
centre  of  gravity.' 

545.  Hence  §§  510,  515  show  that — 

{a)  The  centre  of  gravity  of  a  centrobaric  body  necessarily  lies,  in  its- 
interior  ;^^x  in  other  words,  r<?«  otily  be  reached  from  external  sface  by 
a  path  cutting  through  some  of  its  niass..^    And 

{b)   No\entrobaric  vody  can  consist  of  parts  isolaiedfrom  one  another y 

»  Thomson.    Proc.  R.  S.  £.,  Feb.  1864. 


STATICS  OF  A  PARTICLE.— ATTRACTION.      197 

each  in  spcue  external  to  all',  in  other  words,  the  order  boundary  of  every 
antrobaric  body  is  a  single  closed  surface. 

Thus  we  see,  by  (a),  that  no  symnietrlcal  ring,  or  hollow  cylinder 
with  open  ends,  can  have  a  centre  of  gravity;  for  its  centre  of 
gravit}',  if  it  had  one,  would  be  in  its  axis,  and  therefore  external  to 
its  jnass. 

546.  If  any  mass  ivhatcver,,  M,  and  any  single  surface,  S,  com' 
pletely  enclosing  it  be  given,  a  distribution  of  any  givcti  amount,  M', 
of  matter  on  this  surface  may  be  found  'i'hich  shall  make  the  li'Jiole 
centrobaric  'djith  its  centre  of  gravity  in  any  given  position  (G)  'with in 
that  surface. 

The  condition  here  to  be  fulfilled  is  to  distribute  M'  over  S,  so  as 
by  it  to  produce  the  potential 

~Tg      ^* 

at  any  point,  £,  of  S;  V  denoting  the  potential  of  M  at  this  point. 
The  possibility  and  singleness  of  the  solution  of  this  problem  were 
stated  above  (§  517).  It  is  to  be  remarked,  however,  that  if  M'  be 
not  given  in  sufficient  amount,  an  extra  quantity  must  be  taken,  but 
neutralized  by  an  equal  quantity  of  negative  matter,  to  constitute  the 
required  distribution  on  S. 

The  case  in  which  there  is  no  given  body  M  to  begin  with  is 
Important]  and  yields  the  following: — 

S17.  A  given  quantity  of  matter  may  be  distributed  in  one  ivay,  but 
in  only  ore  loay,  over  any  given  closed  surface,  so  as  to  constitute  a 
centrobaric  body  toith  its  centre  of  gravity  at  any  given  point  zvithin  it. 

Thus  we  have  already  seen  that  the  condition  is  fulfilled  by  making 
the  density  inversely  as  the  distance  from  the  given  point,  if  the 
surface  be  spherical.  From  what  was  proved  in  §§  519,  524  above, 
it  appears  also  that  a  centrobaric  shell  may  be  made  of  either  half  of 
the  lemniscate  in  the  diagram  of  §  526,  or  of  any  of  the  ovals  within 
it,  by  distributing  matter  with  density  proportional  to  the  resultant 
force  of  7-7  at  /and  ?//  at  /';  and  that  the  one  of  these  points  which 
is  within  it  is  its  centre  of  gravity.  And  generally,  by  drawing  the 
equipotential  surfaces  relatively  to  a  mass  m  collected  at  a  point  /, 
and  any  other  distribution  of  matter  whatever  not  surrounding  this 
point;  and  by  taking  one  of  these  surfaces  which  encloses  /but 
no  other  part  of  the  mass,  we  learn,  by  Green's  general  theorem, 
and  the  special  proposition  of  §  524,  how  to  distribute  matter 
over  it  so  as  to  make  it  a  centrobaric  shell  with  /  for  centre  of 
gravity. 

.548.  Under  hydrokimtics  the  same  problem  will  be  solved  for  a 
cube,  or  a  rectangular  parallelepiped  in  general,  in  terms  of  con- 
verging series;  and  under  electricity  (jn  a  subsequent  volume)  it  will 
be  solved  inf.nite  algebraic  terms  for  the  surface  of  a  lens  bounded 
by  two  spherical  surfaces  cutting  one  another  at  any  sub-multiple  of 
two  right  angles,  and  for  either  part  obtained  by  dividing  this  surface 


198  ABSTRACT  DYNAMICS. 

in  two  by  a  third  spherical  surface  cutting  each  of  its  sides  at  right 
angles. 

549.  Matter  may  be  distributed  in  an  infinite  ninnber  of  mays 
throughout  a  given  closed  space,  to  constitute  a  centrobaric  body  with  its 
centre  of  gravity  at  any  given  point  7ijit7iin  it. 

For  by  an  infinite  number  of  surfaces,  each  enclosing  the  given 
point,  the  whole  space  between  this  point  and  the  given  closed  surface 
may  be  divided  into  infinitely  thin  shells;  and  matter  may  be  dis- 
tributed on  each  of  these  so  as  to  make  it  centrobaric  with  its  centre 
of  gravity  at  the  given  point.  Both  the  forms  of  these  shells  and  the 
quantities  of  matter  distributed  on  them,  may  be  arbitrarily  varied  in 
an  infinite  variety  of  ways. 

Thus,  for  example,  if  the  given  closed  surface  be  the  pointed  oval 
constituted  by  either  half  of  the  lemniscate  of  th*e  diagram  of  §  526, 
and  if  the  given  point  be  the  point  /  within  it,  a  centrobaric  solid 
may  be  built  up  of  the  interior  ovals  with  matter  distributed  over 
them  to  make  them  centrobaric  shells  as  above  (§  547).  From  what 
was  proved  in  §  534,  we  see  that  a  solid  sphere  with  its  density 
varying  inversely  as  the  fifth  power  of  the  distance  from  an  external 
point,  is  centrobaric,  and  that  its  centre  of  gravity  is  the  image  (§  530) 
of  this  point  relatively  to  its  surface. 

550.  The  centre  of  gravity  of  a  centrobaric  body  composed  of 
tnie  gravitating  matter  is  its  centre  of  inertia.  For  a  centrobaric 
body,  if  attracted  only  by  another  infinitely  distant  body,  or  by  matter 
so  distributed  round  itself  as  to  produce  (§  517)  uniform  force  in 
parallel  lines  throughout  the  space  occupied  by  it,  experiences  (§  544) 
a  resultant  force  always  through  its  centre  of  gravity.  But  in  this 
case  this  force  is  the  resultant  of  parallel  forces  on  all  the  particles  of 
the  body,  which  (see  Properties  of  Matter,  below)  are  rigorously  pro- 
portional to  their  masses:  and  it  is  proved  that  the  resultant  qf 
such  a  system  of  parallel  forces  passes  through  the  point  defined  in 
§  195,  as  the  centre  of  inertia. 

551.  The  moments  of  inertia  of  a  centrobaric  body  are  equal 
round  all  axes  through  its  centre  of  inertia.  In  other  words  (§  239), 
all  tiiese  axes  are  principal  axes,  and  the  body  is  kinetically  sym- 
Tietrical  round  its  centre  of  inertia. 


CHAPTER  VII. 


STATICS   OF   SOLIDS   AND    FLUIDS. 


552.  Forces  Avhose  lines  meet.     Let  ABC  be  a  rigid  body  acted 
on   by   two   forces,    P  and    Q,   applied    to    it 
at   ditlerent   points,  D  and  J£  respectively,  in 
lines  in  the  same  plane. 

Since  tiie  lines  are  not  parallel,  they  will 
meet  if  produced;  let  them  be  produced  and 
meet  in  O.  Transmit  the  forces  to  act  on 
that  point;  and  the  result  is  that  we  have 
simply  the  case  of  two  forces  acting  on  a 
material  point,  which  has  been  already  con- 
sidered. 

553.  The  preceding  solution  is  applicable  to  every  case  of  non- 
parallel  forces  in  a  plane,  however  far  removed  the  point  may  be  in 
which  their  lines  of  action  meet,  and  the  resultant  will  of  course  be 
found  by  the  parallelogram  .of  forces.  The  limiting  case  of  parallel 
forces,  or  forces  whose  lines  of  action,  hov/ever  far  produced,  do  not 
meet,  was  considered  above,  and  the  position  and  magnitude  of 
the  resultant  were  investigated.  The  following  is  an  independent 
demonstration  of  the  conclusion  arrived  at. 

554.  Parallel  forces  in  a  plane.  The  resultant  of  two  parallel 
forces  is  equal  to  their  sum,  and  is  in  the  parallel  line  which  divides 
any  line  drawn  across  their  lines  of  action  into  parts  inversely  as  their 
magnitudes. 

1°.  Let  F  and  Q  be  two  parallel  forces  acting  on  a  rigid  body  in 
similar  directions  in  lines  AB  and  CD.  Draw  any  line  AC  across 
their  lines.  In  it  introduce  any 
pair  of  balancing  forces,  S'mAG 
and  6"  in  CH.  These  forces  will 
not  disturb  the  equilibrium  of 
the  body.  Suppose  the  forces 
F  and  S  in  A  G,  and  Q  and  S  in 
CH,  to  act  respectively  on  the 
points  A  and  C  of  the  rigid  body. 
The  forces  F  and  S,  in  AB  and 
AGy  have  ^  single  resultant  in 
some  line  AM,  within  the  angle 


200  ;  ABSTRACT  DYNAMICS. 

GAB;  and  Q  and  S  in  CD  and  CH  have  a  resultant  in  some  line 
CN,  within  the  angle  DCH. 

The  angles  MAC,  NCA  are  together  greater  than  two"  right 
angles,  hence  the  Hnes  MA,  NC  will  meet  if  produced.  Let  them 
meet  in  O.  Now  the  two  forces  P  and  S  may  be  transferred  to 
parallel  lines  through  O.  Similarly  the  forces  Q  and  S  may  be  also 
transferred.  Then  there  are  four  forces  acting  on  O,  two  of  which, 
S  in  OK  ,and  S  in  OL,  are  equal  and  directly  opposed.  They  may, 
therefore,  be  removs-d,  and  t^ere  are  left  two  forces  equal  to  P  and 
Q  in  one  line  on  O,  which  are  equivalent  to  a  single  force  P-v  Qva 
the  same  line. 

2°.  If,  for  a  momeit,  we  suppose  OE  to  represent  the  force  Pf. 
then  the  force  repiesvinting  6"  must  1  e  equal  and  parallel  to  EA,  since 
the  resultant  of  the  two  is  ni  the  direction  OA.     That  is  to  say, 

S:P::EA:  OE; 

and  in  like  manner,  by  considering  the  forces  ^S"  in  OL  and  Q  in  OE, 
we  find  that 

Q'.S::  OE;EC 

Compounding  these  analogies,  we  get  at  once 

Q.P.-.EA  :EC, 

that  is,  the  parts  into  which  the  line  is  divided  by  the  resultant  are 
inversely  as 'the  forces. 

555.  Forces  in  dissimilar  directions.  The  resultant  of  two  parallel 
forces  in  dissimilar  directions',  of  which  one  is  greater  than  the  other, 
is  found  by  the  following  rule  :  Draw  any  line  across  the  lines  of  the 
forces  and  produce  it  across  the  line  of  the  greater,  until  the  whole 
line  is  to  the  part  produced  as  the  greater  force  is  to  the  less ;  a  force 
equal  to  the  excess  of  the  greater  force  above  the  less,  applied  at  the 
extremity  of  this  line  in  a  parallel  line  and  in  the  direction  similar  to 
that  of  the  greater,  is  the  resultant  of  the  system. 

Let  P  and  Q  in  KK'  and  LL',  be  the 
contrary  forces.  From  any  point  A,  in 
the  line  of  P,  draw  a  line  AB  across  the 
line  of  Q  cutting  it  in  B,  and  produce  the 
line  to  E,  so  that  AE  :  BE  ::  Q -.  P. 
Through  E  draw  a  line  MM  parallel  to 
^  M  KK'oxLL'. 

In  MM'  introduce  a  pair  of  balancing  forces  each  equal  to  Q-P- 
Then  P in  AK'  and  Q-P  in  EM  have  a  resultant  equal  to  their 

^  In  future  the  word  'contrary'  will  be  employed  instead  of  the  phrase 
'parallel  and  in  dissimilar  directions'  to  designate  merely  diruticnal  oppcsiiwn, 
while  the  unqualified  word  'opposite'  will  be  understood  to  signify  contrary  and\ 
in  one  lint. 


STATICS   OF  SOLIDS  AND  FLUIDS.  20 r 

sum,  or  (2-     This  resuftant  is  in  the  line  LL ;  for,  from  the  ana- 
logy, 

AE:BE::  Q  :  P, 

we  have  AE- BE  :  BE  ::  Q~  P  :  P, 

or  AB  -.BE  ::  Q-P:P. 

Hence  P  in  AK',  Q  in  BL',  and  Q-  P'm  EM  are  in  equilibrium 
and  may  be  removed.  There  remains  only  Q-Pm  EM',  v.hich  is 
therefore  the  resultant  of  the  two  given'forces,  This  fails  when  the 
forces  are  equal. 

556.  Any  number  of  parallel  forces  in  a  plane.  Let  P^,  P^,  P^, 
etc.,  be  any  number  of  parallel  forces 
acting  on  a  rigid  body  in  one  plane. 
To  find  their  resultant  in  position  and 
magnitude,  draw  any  line  across  their 
lines  of  action,  cutting  them  in  points, 
denoted  respectively  by  A^,  A^,  A^,  etc., 
and  in  it  choose  a  point  of  reference  O.  Let  the  distances  of  the 
lines  of  the  forces  from  this  point  be  denoted  by  ^  ,  a.,  a  ,  etc.:  as 
OA,  =  a„  OA,  =  a,,  etc.  Also  let  P  denote  the  resultant,' and  .r  its 
distance  from  O. 

Find  the  resu'tant  of  any  two  of  the  forces,  as  /\  and  P^,  by 
§  554,     Then  if  we  denote  this  resultant  by  P',  we  have 

F'  =  P,-^P^. 

Di\ ide  Ai  A,  in  E'  into  parts  inversely  as  the  forces,  so  that 

P,-<A,E  =  P^:<E'A,^. 

Hence  if  we  denote  OE'  by  x'  we  have 

Px{x'-a,)  =  P^x{i7^-x') 

or  (P,  +  P^x■  =  P,c7,  +  P^a^, 

that  is  P' x'  =  P^a,-i-  P^ a,^. 

Similarly  we  shall  find  the  resultant  of  P'  and  P^  to  be 

P"  =  P'^-P^  =  P,  +  P^-^-P^; 

and  P"  x"  =  Rx'  +  P.,a^=  P,  a,  +  P^  (7,  +  P,a^. 

Ht-'nce,  finally  we  have 

P  =  P,^P^  +  P^+ ~P„ (I) 

and  Px  =  P,a^  +  P,a^  +  P^a^+ - -^„  ^,. (^) 

In  this  method  negative  forces  or  negative  values  of  any  of  the 
quantities  ^1,  ^j,  ...,  may  be  included,  provided  the  generahzed  rules  of 
multiplication  and  addition  in  algebra  are  followed. 

557.  Any  number  of  parallel  forces  not  in  one  plane.  To  find  the 
resultant,  let  a  plane  cut  the  lines  of  all  the  forces,  and  let  the  points 


ABSTRACT  DYNAMICS. 


in  which  they  are  cut  be  specified  by  reference  to  two  rectangular 
axes  in  the  plane.  Let  the  plane  be  YOX:  OX,  OY,  the  axes  of 
reference,  O  the  origin  of  co-ordinates,  and  A^,  //,,  A^,  etc.,  the  points 
in  which  the  plane  cuts  the  lines  of  the  forces,  P,,  P,,,  F^,  etc  Thus 
each  of  these  points  will  be  specified  by  perpendiculars  drawn  from 

it  to  the  axis.  Let  the  co-ordinates 
of  the  point  A.^  be  denoted  by  x^  y^y 
ofy/^,  by  ,r„  j'^;  and  so  on;  that  is, 
ON,  =  X, ,NA,  =;-, ;  ON^  =  x^,N^  A^ 
=y^,  etc.;  let  also  the  final  resultant  be 
denoted  by  H,  and  its  co-ordinates  by 
X  and  y. 

Find  the  resultant  of  F,  and  F^  by 

joining  A,  A„,  and  dividing  the   line 

-X       inversely  as   the   forces.     Suppose  E' 

the  point  in  which  this  resultant  cuts 


4 

E 

/ 

K 

A 

1\ 


N, 


j\'N 

the  plane  of  reference.     Then 

F,  X  A.F'  =  F.,^  E'A:,. 

To  find  the  co-ordinates,  which  may  be  denoted  by  x'y',  of  the 
point  E'  with  reference  to  OX  and  OY;  draw  E'N  perpendicular 
to  OX  and  cutting  it  in  N',  and  from  A,  draw  ^,  A"  parallel  to  OX, 
or  perpendicular  to  A„N,„  and  cutting  it  in  AT  and  EN'  in  M. 
Then  (Euclid  vi.  2) 

A,E'  \E'A.^  ■.•.A.M-.MK. 

Hence  F,  x  A,M^  P.^  x  MK, 

or  F,{x'-x^  =  F„_{x^-x% 

whence  we  get  (/*i  4-  F^  x  =  F^x,  -f  F„x, ; 

and  since  K' =  F,+ F„, 

we  have  F'x'  =  F^x,  +  T.x.,, 

and  similarly,  F'y'  -  F^y,  +  F„y.,. 

We  may  find  the  resultant  of  F'  and  F^  in  like  manner,  and  so 
with  all  the  forces.     Hence  we  have  for  the  final  resultant, 

F  =  F,  +  F^  +  F^+  +F^ (3), 

Fx  =  F,x,  +  F._,x„^^F.^x^+  ...    +F,x^ (4), 

Fy  =  F,y,  +  F.^y^_-i-F^^y^4....    +  Fj,, (5). 

These  equations  may  include  negative  forces,  or  negative  co- 
ordinates. 

558.'  Conditions  of  equilibrium  of  any  nunilier  of  parallel  forces. 
In  order  that  any  given  parallel  forces  may  be  in  equilibrium,  it  is 
not  sufficient  alone,  that  their  algebraic  sum  be  equal  to  zero. 

For,  let  F  =  F  +  F,  +  etc.  =  o. 


STATICS   OF  SOL/DS  AXD   FLUIDS  203 

From  this  equation  it  follows  that  if  the  forces  be  divided  into  two 
groups,  one  consisting  of  the  forces  reckoned  poiitive,  the  other  of 
those  reckoned  negative,  the  sum,  or  resultant  (j'  556),  of  the  former 
is  equal  to  the  resultant  of  the  latter;  that  is,  ii  R  and  'A*  denote  the 
resultants  of  the  positive  and  negative  groups  respectively, 

^R  =  'R. 

But  unless  these  resultants  are  directly  opposed  they  do  not  balance 
one  another;  wherefore,  if  {x^y)  and  (x'y)  be  the  co-ordinates  of 
^  and  'R  respectively,  we  must  have  for  equilibrium 

and  ,y  =  'y ; 

whence  we  get  ,R  x  -  'R'x  =  o 

and  ^R  J'  -'R'y  =  o. 

But  ^  ^x  is  equal  to  the  sum  of  those  of  the  terms  F^x^,  Fx,,  etc., 
which  are  positive,  and  'R'x  is  equal  to  the  sum  of  the  others  each 
with  its  sign  changed  :  and  so  for  ^^y  and  'Ry.  Hence  the  pre- 
ceding equations  are  equivalent  to 

F,x^  +  Fji-^  +  -f-  J\x^  =  o. 

Fo\  +  F,y,4. +  F,j;,  =  o. 

We  conclude  that,  for  equilibrium,  it  is  necessary  and  sufficient  that 
each  of  the  following  three  equations  be  satisfied  :  — 

F,  +  F,  +  F^+  ■^F„  =  o (6), 

F,x,  +  F,-x\+  F^x^+ +F^x^  =  o (7), 

F,y,  +  F,y,  +  Fj'^+ +F,^y^  =  o (S). 

559.  If  equation  (6)  do  not  hold,  but  equations  (7)  and  (8)  do,  the 
forces  have  a  single  resultant  through  the  origin  of  co-ordinates.  If 
ecjuation  (6)  and  either  of  the  other  two  do  not  hold,  there  will  be  a 
single  resultant  in  a  line  through  the  corresjjonding  axis  of  reference. 
the  co-ordinates  of  the  other  vanishing.  If  equation  (6)  and  either  of 
the  other  two  do  hold,  the  system  is  reducible  to  a  single  couple  in  a 
plane  through  that  line  of  reference  for  which  the  sum  of  the  products 
is  not  equal  to  nothing.  If  the  plane  of  reference  is  jjerpendicular  to 
the  lines  of  the  forces,  the  inoment  of  this  couple  is  equal  to  the  sum 
of  the  products  not  ccjual  to  nothing. 

560.  In  finding  the  resultant  of  two  contrary  forces  in  any  case  in 
which  the  forces  are  unequal — the  smaller  the  ditierence  of  magnituile 
between  them,  the  farther  removed  is  the  point  of  application  of  the 
resultant.  When  the  difference  is  nothing,  the  point  is  removed  ta 
an  infinite  distance,  and  the  construction  (,^  555)  is  thus  rendered 
nugatory.  The  general  solution  gives  in  this  case  R  ~  o  ;  yet  the 
forces  are  not  in  equilibrium",  since  they  are  not  directly  oppo-ed. 
Hence  two  equal  contrary  forces  neither  balance,  nor  have  a  iingje 
resultant.     It  is  clear  that  they  have  a  tendency  to  turn  the  body  to 


204  ABSTRA  CT  D  YNAMICS'. 

which  they  are  applied.    This  system  was  by  Poinsot  denominated 
a  couple. 

lu  actual  cases  the  direction  of  a  couple  is  generally  reckoned 
positive  if  the  couple  tends  to  turn  contrary  to  the  hands  of  a  watch 
as  seen  by  a  person  looking  at  its  face,  negative  when  it  tends  to 
turn  with  the  hands.  Hence  the  axis,  which  may  be  taken  to  repre- 
sent a  couple,  will  show,  if  drawn  according  to  the  rule  given  in  §  201, 
.whether  the  couple  is  positive  or  negative,  according  to  the  side  of  its 
plane  from  which  it  is  regarded. 

561.  Proposition  I.  Any  two  couples  in  the  same  or  in  parallel 
planes  are  in  equilibrium  if  their  moments  are  equal  and  they  tend  to 
turn  in  contrary  directions. 

1°.  Let  the  forces  of  the  first  couple  be  parallel  to  those  of  the 
second,  and  let  all  four  forces  be  in  one  plane. 

Let  the  forces  of  the  first  couple  be 
P  in  AB  and   CD,  and  of  the  second 
P'  in  A'B'  and   CD'.     Draw  any  line 
EF'  across  the  lines  of  the  forces,  cut- 
ting them  respectively  in  points  E,  E, 
E'  and    F' ;   then   the  moment  of  the 
B       D  U   B         D'       first  couple  is /".isi^ and  of  the  second 
P'.  E'E' ;  and  since  the  moments  are  equal  we  have 
P.  EE=^P'.  E'E'. 
Of  the  four  forces,  P  in  AB  and  P'  in  CD  act  in  similar  direc- 
tions, and  P  ill  CD  and  P'  in  A'B'  also  act  in  similar  directions; 
and  their  resultants  respectively  can  be  determined  by  the  general 
method  (§  556).     The  resultant  of  P  in  AB,  and  P'  in  CD,  is  thus 
found  to  be  .equal  to  P+  P' ,  and  if  HL  is  the  line  in  which  it  acts, 

P.EK=P'.KF'. 
Again,  we  have  P-  EF=  P'.  E'F': 

Subtract  the  first  member  of  the  latter  equation  from  the  first 
member  of  the  former,  and  the  second  member  of  the  latter  from  the 
second  member  of  the  former:  there  remains 

p.  FK=  P'.  KE, 
from  which  we  conclude,  that  the  resultant  of  P  in  CD  and  P'  in 
A'B'  is  in  the  line  LH.  Its  magnitude  is  /*+  P' .  Thus  the  given 
system  is  reduced  to  two  equal  resultants  acting  in  opposite  directions 
in  the  same  straight  line.  These  balance  one  another,  and  therefore 
the  given  system  is  in  equilibrium. 

Corollary.  A  couple  may  be  transferred  from  its  own  arm  to  any 
other  arm  in  the  same  line,  if  its  moment  be  not  altered. 

562.  Proposition  I.  2".  All  four  forces  in  one  plane,  but  those  of 
one  couple  not  parallel  to  those  of  the  other. 

Produce  their  fines  to  meet  in  four  points;  and  consider  the  paral- 
lelogram thus  formed.  The  products  of  tlie  sides,  each  into  its  per- 
pendicular distance  from  the  side  parallel  to  it,  are  equal,  each  product 


STATICS  OF  SOLIDS  AXD  FLUIDS.  205 

being  the  area  of  the  parallelogram.  Hence,  since  the  moments  of  the 
two  couples  are  equal,  their  forces  are  proportional  to  the  sides  of  the 
parallelogram  along  which  they  act.  And,  since  the  coupks  tend  to 
turn  in  opposite  directions,  the  four  forces  represented  b\»the  sides  of 
a  parallelogram  act  in  similar  directions  relatively  to  tne  angles,  and 
dissimilar  directions  in  the  parallels,  and  therefore  balance  one 
another. 

Corollary.  The. statical  effect  of  a  couple  is  not  altered,  if  its  arm 
be  turned  round  any  point  in  the  plane  of  the  couple 

563.  Proposition  I.  3°.  The  two  couples  not  in  the  same  plane, 
but  the  forces  equal  and  parallel. 

Let  there  be   two   couples,   acting  re»         -^f  p 

spectively  on  arms  EF  and  E F' ,  which  £^.         j  .j" 

are   parallel  but  not  in  the  same   plane.  '.'.'■'■■^:'. ^  jp, 

Join   EF'  and  E' F.     These  lines  bisect    Ei* I     '^',"^F 

one  another  in  O.  \p  -^ 

Of  the  four  forces,  P  on.  F  and  P"  on 
E  act  in  similar  directions,  and  their  resultant,  equal  to  P-^  P',  may 
be  substituted  for  them.  It  acts  in  a  parallel  line  through  0.  Simi- 
larly P  on  E  and  P'  on  F'  have  also  a  resultant  equal  to  P  +  P' 
through  O;  but  these  resultants  being  equal  and  opposite,  balance, 
and  therefore  the  given  system  is  in  equilibrium. 

Remark  i. — A  corresponding  dem.onstration  may  be  applied  to 
every  case. of  two  couples,  the  moments  of  which  are  equal,  though 
the  forces  and  arms  may  be  unequal.'  When  the  forces  and  arms  are 
unequal,  the  lines  EF'^E'F CMi  one  another  in  0  into  pans  inversely 
as  the  forces. 

Remark  2. — Hence  as  an  c.\.treme  case,  Proposition  I,  1°,  may  be 
brought  under  this  head.  Let  EFhc  the  arm  of  one  couple,  EF'' 
of  the  other,  both  in  one  straight  line.  Join  F£',  and  divide  it 
inversely  as  the  forces.  Then  FK  \  KE'  ::  EF :  EF'  and  EF'  is 
divided  in  the  same  ratio. 

Corollary.  Transposition  of  couples.  Any  two  couples  in  the 
same  or  in  parallel  planes,  are  equivalent,  provided  their  moments 
are  equal,  and  they  tend  to  turn  in  similar  directions. 

564.  Proposition  H.  Any  number  of  couples  in  the  samiC  or  in 
parallel   planes,    may  be   reduced    to   a  single 

resultant  couple,  whose  moment  is  equal  to  the         .  L, 

algebraic  sum   of  their,  moments,    and    whose  V* 

plane  is  parallel  to  their  planes.  4^ 

Reduce  all   the  couples  to  forces  acting  on  i/^ 

one   arm    AB,  which  may  be    denoted   by  a.        A^  ^If 

Then  \i  P^,  I\,  P..  etc.,  be  the  forces,  the  mo-  i7]_ 

ments   of  the    couples  will  be  P.a,  P.a,   P,a,  p 

etc.     Thus  we  ha\e   P,,  P.,  P^,  etc.,  in  AK,  ,j 

reducible    to   a    siiv_:Ic    force,    tlicir    sum,    and  '1- 

similarly,  a  single  force  /^j  + /",  +  etc.,  in  PL. 


'B' 


206  ABSTRACT  DYNAMICS, 

These  two  forces  constitute  a  couple  whose  moment  is  (P^  +  P, 
+  Pj+  etc.)  a.  But  this  product  is  equal  to  P^a  +  P^a  +  P^a  +  etc., 
the  sum  of  the  moments  of  the  given  couples,  and  therefore  any 
number  of  couples,  etc.  If  any  of  the  couples  act  in  the  direction 
opposite  to  that  reckoned  positive,  their  moments  must  be  reckoned 
as  negative  in  the  sum. 

565.  Proposition  III.  Any  two  couples  not  in  parallel  planes 
may  be  reduced  to  a  single  resultant  couple,  whose  axis  is  the 
diagonal  through  the  point  of  reference  of  the  parallelogram  de- 
scribed upon  their  axes. 

1°.     Let  the  planes  of  the  two  couples  cut  the  plane  of  the  diagram 

perpendicularly  in  the  lines  AA'  and 
£B'  respectively;  let  the  planes  of 
the  "couples  also  cut  each  other  in  a 
line  cutting  the  plane  of  the  diagram 
in  O.  Through  O,  as  a  point  of  re- 
ference, draw  6>Arthe  axis  of  the  first 
couple,  and  OL  the  axis  of  the  se- 
cond. On  CiTand  OL  construct  the 
parallelogram  OKML.  Its  diagonal 
OM  is  the  axis  of  the  resultant  couple. 
Let  tlie  moment  of  the  couple  acting  in  the  plane  BB',  be  denoted 
by  G,  and  of  tliat  in  AA\  by  H.  For  the  given  couples,  substitute 
two  others,  with  arms  equal  respectively  to  G  and  ZT,  and  therefore 
with  forces  equal  to  unity. 

From  OB  and  OA  measure  off  OE  -  G,  and  0F=  II,  and  let  these 
lines  be  taken  as  the  arms  of  the  two  couples  respectively.  The 
forces  of  the  couples  will  thus  be  perpendicular  to  the  plane  of  the 
diagram:  those  of  the  first,  acting  outwards  at  E,  and  inwards  at  O; 
and  those  of  the  second,  outwards  at  O,  and  inwards  at  F.  Thus,  of 
the  four  equal  forces  which  we  have  in  all,  there  are  two  equal  and 
opposite  at  O,  which  therefore  balance  one  another,  and  may  be 
removed;  and  tliere  remain  two  equal  parallel  forces,  one  acting', 
outwards  at  E,  and  the  other  inwards  at  P,  which  constitute  a  couple 
■on  an  arm  EF. 

This  single  couple  is  therefore  equivalent  to  the  two  given  couples. 

2°.  It  remains  to  be  proved  that  its  axis  is  OM.  Join  EF.  As, 
by  construction,  OL  and  OK  arc  respectively  perpendicular  to  OA, 
and  OJJ,  tlie  angle  XOL  is  equal  to  the  angle  A  OB'.  Hence,  MLO 
the  supplement  of  the  former  is  equal  to  EOF,  the  supplement  of  tlie 
latter.  Cut  OK  is  equal  to  OE ;  each  being  equal  to  the  moment  of 
the  first  of  the  given  couples ;  and  therefore  L3f,  which  is  equal  to 
the  former,  is  equal  to  OE.  Similarly  OL  is  equal  to  OF.  Thus  there 
are  two  triangles,  AILO  and  EOF,  with  tv/o  sides  of  one  respectively 
equal  to  two  sides  of  the  otiicr,  and  the  contained  angles  equal:  there- 
fore the  remaining  sides  OM,  EF  are  equal,  and  the  angles  LOM, 
OFF  are  equal.     But  since   OL  is  perpendicular  to  OF,  OM  is 


STATICS.  OF  SOLIDS  AND  FLUIDS.  207 

perpendicular   to   EF.     Hence    OM  is   the   axis   of  the   resultant 
couple. 

566.  Proposition  IV.  Any  number  of  couples  whatever  are  cither 
in  equilibrium  with  one  another,  or  may  be  reduced  to  a  single  couple, 
under  precisely  the  same  conditions  as  those  already  in\e3tigated  for 
forces  acting  on  one  point,  the  axes  of  the  couples  being  now  taken 
ever)-^vhere  instead  of  the  lines  formerly  used  to  represent  the  forces: 

1°.     Resolve  each  couple  into  three  components  having  their  axes 

along  three  rectangular  lines  of  reference,  OX,  OY,  OZ.     Add^all 

the  components  corresponding  to  each  of  these  three  lines.     Then 

if  the   resultant   of  all   the   couples  whose  axes  are  along  the  line 

OX,  be  denoted  by  Z, 

OY,      „        „        JI, 

OZ,       „        „         IV, 

and  if  G  be  the  resultant  of  these  three,  we  have 

G=  ^f{r  +  JlP  +  N'): 

and  if  t,  7],  6,  be  the  angles  which  the  axis  of  this  couple  C,  rriakes 
with  the  three  axes  OX,  0  Y,  OZ,  respectively,  we  have 

COS^=T-,;  C0S7J  =  -^;  cos&  =  ^. 

567.  2'.  Conditions  of  equilibrium  of  any  number  of  couples.  For 
equilibrium  the  resultant  couple  must  be  equal  to  nothing:  but  as  it  is 
compounded  of  three  subsidiary  resultant  couples  in  planes  at  right 
angles  to  one  another,  they  also  must  each  be  equal  to  nothing.  The 
remarks-already  made,  and  the  equations  already  given  in  §§  471,  472, 
apply  with  the  necessary  modification  to  couples  also.  Thus,  for 
instance,  the  equations  of  equilibrium  are 

G^  cos  ^,  +  G^  cos  ^,  +  G^  cos  ^3  +  etc.  =  o, 

(7,  cost;,  +  G^  cos  ^a  +  G^  cos  t]^  +  etc.  =  o, 

Cj  cos  Bx  +  Gt  cos  6t  +  G^  cos  6^  +  etc.  -  o. 

568.  Before  investigating  the  conditions  of  equilibrium  of  any 
number  of  forces  acting  on  a  rigid  body,  we  shall  establish  some 
preliminary  propositions. 

1*.  A  force  and  a  couple  in  the  same  or  in  parallel  planes  may  be 
reduced  to  a  single  force.  Let  the  plane  of  the  couple  be  the  plane 
of  the  diagram)  and  let  its  moment  be 
denoted    by    G.      Let    F,   acting    in    the  ^ 

line   OA  in  the  same  plane,  be  the  force.  /  /        \^ 

Transfer  the  couple  to  an  arm  (which  may 
be  denoted  by  a)  through  the  point  0,  such 
that  each  force  shall  be  equal  to  R',  and  let 
its'  position  be  so  chosen,  that  one  of  the 
forces  shall  act  in  the  same  straight  line  with 
R  in  OA,  but  in  the  opposite  direction  to  it. 


2o<i  ABSTRACT  DYNAMICS. 

R  and  G  being  known,  the  length  of  this  arm  can  be  found,  for 
since  the  moment  of  the  transposed  couple  is 

■we  have  a  -  -^, 

Through  <9then,  draw  a  line  <96>' perpendicular  to  OA,  making  it 
equal  to  a.  On  this  arm  apply  the  couple,  a  force  equal  to  J?,  acting 
on  O'  in  a  line  perpendicular  to  OO,  and  another  in  the  opposite 
direction  at  the  other  extremity.  There  are  now  three  forces,  two  of 
which,  being  equal  and  opposite  to  one  another,  in  the  line  AA\  may 
be  removed.  One,  acting  on  the  point  (7,  remains,  which  is  there^ 
fore  equivalent  to  the  given  system. 

569.  2".  A  couple  and  a  force  in  a  given  line  inclined  to  its  plane 
may  be  reduced  to  a  smaller  couple  in  a  plane  perpendicular  to  the 
force,  and  a  force  equal  and  parallel  to  the  given  force. 

Let  OA  be  the  line  of  action  of  the  force  R, 
and  let  6>^be  the  axis  of  the  couple.  Let 
the  moment  be  denoted  by  G  :  and  let  A  OK^ 
the  inclination  of  its  axis  to  the  line  of  thef 
force,  be  0.  Drav/  OB  perpendicular  to  OA. 
By  Prop.  IV.  (§  566)  resolve  the  couple  into 
two  components,  one  acting  round  OA  as- 
O  axis,  and  one  round  OB.     Thus  the  compo- 

nent round  OA  will  be 

G  cos  0^ 
and  the  component  round  OB, 

G  sin  e. 

.Now  as  G  sin  Q  acts  in  the  same  plane  as  the  given  force  R,  this  com- 
ponent together  with  R  may  be  reduced  by  §  568  to  one  force.  This 
force  which  is  equal  to  R,  will  act  not  at  O  in  the  line  OA^  but  in  a 
parallel  line  through  a  point  O'  out  of  the  plane  of  the  diagram.  Thus 
the  given  system  is  reduced  to  a  smaller  couple  G  cos  Q,  and  to  a 
force  in  a  line  which,  by  Poinsot,  was  denominated  the  central  axis  df 
the  system. 

570.  3°.  Any  number  of  forces  may  be  reduced  to  a  force  and 
a  couple. 

Let  -Pj  acting  on  M^  be  one  of  a  number  of  forces  acting  in 
different  directions  on  different  points  of  a  rigid 
body.  Choose  any  point  of  reference  6>,  for  the 
dirferent  forces,  and  tluough  it  draw  a  line  AA' 
parallel  to  the  line  of  the  first  force  P^.  Tlu-ough  <?, 
draw  OO'  perpendicular  to  A  A  or  the  line  of  the 
force  Fy.  In  the  line  A  A'  introduce  two  equal  op- 
posite forces,  each  equal  to  P-^^  There  are  now 
three  forces,  producing  the  same  effect  as  the  given 
force,  and  they  may  be  grouped  differently :  P^  acting 


STATICS  OF  SOLIDS  AND  FLUIDS  209 

!n  0  in  the  line  OA,  and  a  couple,  /\  acting  at  O",  and  /\  at  0  in 
the  line  OA,  on  an  arm  00'.  Reduce  similarly  all  the  other  forces, 
each  to  a  force  acting  on  O,  and  to  a  couple.  But  all  the  couples 
thus  obtained  are  equivalent  to  a  single  couple,  and  all  the  forces  are 
equivalent  to  one  force.     Hence,  &c. 

571.  Reduction  of  any  number  of  forces  to  their  simplest  equi 
valent  system. 

Suppose  any  number  of  forces  acting  in  any  directions  on  different 
points  of  a  rigid  body.  Choose  three  rectangular  planes  of  reference 
meeting  in  a  point  O,  the  origin  of  co-ordinates.  In  order  to  effect 
the  reduction  it  is  necessary  to  bring  in  all  the  forces  to  the  point  0. 
This  may  be  done  in  two  different  ways — either  in  two  steps,  or 
directly. 

672.  I*  Let  the  magnitudes  of  the  forces  be  /i,  /*,,  &c.,  and  the 
co-ordinates,  with  reference  to  the  rectangular  planes,  of  the  points  at 
which  they  act  respectively,  be  (^1,^1,  sj,  (^a,^i,  ^s)>  ^c.  Let  also 
the  direction  cosines  be  (^,  nt^,  n^,  {/,,  m^,  «,),  &c.  Resolve  each 
force  into  three  components,  parallel  to  OX,  OV,  OZ,  respectively. 
Thus,  if  (Xi,  Viy  Zj),  &c.,  be  the  components  of  F^,  See,  we  shall 
iiave 

X,  =  F,/^X,  =  A/,;&:c  (i) 

y.^F^mr,  Y.^F.m,;  (kc,  (2) 

^  =  i>«i;^  =  i'.«,;&a  (3) 

To  transfer  these  components  to  the  point  0,  Let  X^,  in  MJC,  be 
the  component,  parallel  to  OX,  of  the  force  /^  acting  on  the  point  M. 
From  M  transmit  it  along  its  line  to  a 
point  JV  in  the  plane  ZO  V:  the  co-or- 
dinates of  this  point  will  be_>'„  Zy  From 
iVdraw  a  perpendicular  NB  to  OV, 
and  through  F  draw  a  line  parallel  to 
MK  ox  OX.  Introducing  in  this  line 
a  pair  of  balancing  forces  each  equal 
to  X^,  we  have  a  couple  acting  on  an 
arm  ^i  in  a  plane  parallel  to  XOZ,  ^~^ 
and  a  single  force  X^  parallel  to  OX  Y^ 
in  the  plane  XO  V.  The  moment  of 
this  couple  is  ^i0i,and  its  axis  is  along 
OY.  Next  transfer  the  force  Xi  from  F  to  0,  by  introducing  a  palj 
of  balancing  forces  in  XOX,  one  of  which,  with  the  force  X^  in  the 
line  through  F  parallel  to  A^'Xand  the  direction  similar  to  OX,  form 
a  couple  acting  on  an  armj'i.  This  couple,  when>',  and  X^  are  both 
positive,  tends  to  turn  in  the  plane  XO  V  from  OVto  OX.  Therefore 
by  the  rule,  §  201,  its  axis  must  be  drawn  from  Oin  the  direction  0Z\ 
Hence  its  moment  is  to  be  reckoned  &s~  X^y^.  Besides  this  couple 
there  remains  a  single  force  equal  to  X^,  in  the  direction  OX,  through 
the  point  O.  Similarly  by  successive  steps  transfer  the  forces  Ki,  .^, 
Voi.  2.:— 10 


a  I  o  ABSTRA  CT  D  YNAMICS. 

to  the  origin  of  co-ordinates.  In  this  way  six  couples  of  transference 
are  got,  three  tending  to  turn  in  one  direction  round  the  axes  respec- 
tively, and  three  in  the  opposite  direction;  and  three  single  forces  at 
right  angles  to  one  another,  acting  at  the  point  O.  Thus  for  the  force 
/*,,  at  the  point  (jf,,>'x,  Zi),  we  have  as  equivalent  to  it  at  the  point 
Ot  three  forces  -Xi,  Fi,  ^,  and  three  couples; 

Zj'j  -  YxW  moment  of  the  couple  round  0X\  (4) 

X^z^-Z^x^^  moment  of  the  couple  round  OK;  (5) 

Y^x^-X^y-Ct  moment  of  the  couple  round  CZ  (6) 

All  the  forces  may  be  brought  in  to  the  origin  of  co-ordinates  in  a 
similar  way. 

573.    2".    Otherwise:  Let  P  be  one  of  the  forces  acting  in  the  line 
rn  MT  on  a  point  Afof  a  rigid  body.  Let 

'  O  be  the  origin  of  co-ordinates;  OX, 

OY,  OZ,  three  rectangular  lines  of  re- 
ference. Join  (9il/and  produce  the  line 
to  S.  From  O  draw  ON,  cutting  at 
right, angles  in  the  point  N,  the  line 
i^^produced  through  M.  Let  ON 
be  denoted  by/,  and  the  angle  TMS 
by  K.  In  a  line  through  O  parallel  to 
MT  (not  shown  in  diagram)  suppose 
introduced  a  pair  of  balancing  forces  each  equal  to  P.  We  have 
thus  a  single  force  equal  to  P  acting  at  0,  and  a  couple,  whose 
moment  is  Pp,  in  the  plane  ONM.  The  direction  cosines  of  this 
plane,  or,  which  is  the  same  thing,  the  direction  cosines  of  a  per- 
pendicular to  it,  that  is,  the  axis  of  the  couple  are  (§  464),  if  we  denote 
them  by  ^,  x,  ^>  respectively, 


</,= 

y      z 

-  «  -  -  VI 

r       r 

sin  K     * 

X  -■ 

Z  .     X 

-/--  n 
r      r 

sin  K    ' 

^ 

X        y  , 
r        r 

sm  K 

No 

IV  in 

the 

triangle 

ONM, 

ON 

=  OMs\x\  OMN^ 

that 

is 

P 

=  r  sin  K. 

STATICS  OF  SOLIDS  AND  FLUIDS  i\i 

Hence,  if  we  substitute  /  for  its  value  in  the  three  preceding  equations, 
the  expression  for  the  direction  cosines  are  reduced  to 

'i'^—p-^  (7) 

Iz-nx  .„. 

x=-^.  (S) 

mx-ly 
^  =  -—^'  (9) 

To  find  the  component  couples  round  OX^  0  Vy  OZ,  multiply  these 
direction  cosines  respectively  by  Pj>;  whence  we  get 

F'/>.<p  =  P{tiy  -  mz)y  moment  of  couple  round  OX^  ( i  o) 

Pp.X=P{lz- nx)y  moment  of  couple  round  OYy  ( 1 1 ) 

Fp ,  \li  =  P  {mx  -  ly),  moment  of  couple  round  OZ.  (12) 

That  this  result  is  the  same  as  that  got  by  the  other  method  will  be 
evident,  by  considering  that  (equations  i,  2,  3), 

Pl^X;    Pm=  K;    /Vi  =  Z 

574.  When  by  either  of  the  methods  all  the  forces  have  been  re- 
ferred to  O,  there  is  obtained  a  set  of  couples  acting  round  OX,  0  V, 
OZ;  and  a  set  of  forces  acting  along  OX,  O  V,  OZ.  Find  then  the 
resultant  moments  of  all  the  couples;  and  the  sums  of  all  the  forces: 
if  Z,  My  N  be  the  resultant  moments  round  OXy  O  V,  OZ  respectively, 
we  have 

Z  =  (Z,  y,  -  F.  z)  +  (Z,y,  -  V,z,)  +  &c.  {13) 

M  =  (^Yj  z^  -  Zi  Xi)  +  (X  Zj  -  Za  x^)  +  &c.  ( 1 4) 

N={Y,x,-X,y,)  +{Y,x,-X,y;)  +  8zc.  (15) 
and  if  Xy  K,  Z  be  the  resultant  forces, 

^  X=X,+X^  +  X^  +  kc  (16) 

*  y=v,+  y,+  y^  +  &c.  (17) 

Z  =  Z^  +  Z^+Z^-h8ic  (18) 

575.  Finally,  find  the  resultant  of  the  three  forces  by  the  formulae 
of  Chap.  VI,  and  the  resultant  of  the  three  couples  by  Prop.  IV 
(§  566).  Thus,  if  /,  m,  n  be  the  direction  cosines  of  the  resultant  force 
Ry  we  have  (g  463,  467) 

,    X  Y  Z  .    , 

and  if  X,  /x,  v  be  the  direction  cosines  of  the  axis  of  the  resultant 
couple,  we  have  (§  566) 

.     Z  M  N  ,    , 

^^G*     ^^G*      "^G-  ^'°^ 


2 1 2  ABSTRA  CT  D  YNAMICS. 

576.  Conditions  of  Equilibrium.  The  conditions  of  equilibrium 
of  three  forces  at  right  angles  to  one  another  have  been  already  stated 
in  §  470;  and  the  conditions  for  three  rectangular  couples  in  §  567. 

If  a  body  be  acted  on  by  three  forces  and  three  couples  simul- 
taneously, all  the  conditions  applicable  v/hen  they  act  separately,  must 
also  be  satisfied  when  they  act  conjointly,  since  a  force  cannot  balance 
a  couple.  Six  Equations  of  Equilibrium  therefore  are  necessary  and 
sufficient  for  a  rigid  body  acted  on  by  any  number  of  forces.  These 
are 

/*!  cos  Oj  +  F^  cos  Cj  +  &c.  =  o, 

Pi  COS^i  +  F^  COSySj  +  &C.  =:  o, 

F^  COS  7,  +  F^  COS  yj  +  &c.  =  o, 
Gi  COS  ^j  +  G^,  COS  4  +  5''c.  =  o, 
G^  COS  T/j  +  G^  cos  y)i  +  &c.  =  0, 
Gi  cos  0^  +  G^  cos  9^  +  &c.  =  o. 

577.  If  the  line  of  the  resultant  found  by  §  575,  is  perpendicular 
to  the  plane  of  the  couple,  that  is,  if 

\  =  l,     fxzzm,     v  =  n', 

L     M    N  ,    . 

or  X=V  =  Z*  ^''^ 

the  system  cannot  be  reduced  to  another  with  a  force  and  a  smaller 
couple,  and  in  this  case  the  line  found  for  the  resultant  force  is  the 
central  axis  of  the  system, 

578.  If,  on  the  other  hand,  the  plane  of  the  couple  is  parallel  to  the 
line  of  the  force,  or  the  axis  of  the  couple  perpendicular  to  the  line  of 
the  force,  that  is,  if 

A  +  7///X  +  nv  =  o, 

or  ZX-hMY+NZ=o,  (22) 

the  force  and  couple  may  (§  56S)  be  reduced  to  one  force :  and  this 

force  is  parallel  to  the  former,  at  a  distance  from  it  equal  to  -^ ,  in 

the  plane  of  it  and  the  couple.  Thus,  X(7  being  the  foot  of  the 
perpendicular  from  the  origin  on  the  line  of  action  of  the  resultant 
force,  6'C>'will  be  perpendicular  to  the  line  of  the  resultant  force,  and 
to  the  axis  of  the  resultant  couple,  and  therefore  its  direction  cosines 

are  (§464,  ^); 

mv  -  njx,     nX  —  /v,     /yx  -  m\,  (23) 

each  of  which  will  be  positive  when  C  lies  within  the  solid  angle 

/J 
edged  by  OX,  OV,  OZ.     Hence,  remembering  that  00'  =  -^.,  and 

using  the  expressions  (19)  and  (sq),  we  find  for  the  co-ordinates  of  C/ 
YN-ZM       ^L-XN       XM-  YL 


STATICS  OF  SOLIDS  AND  FLUIDS  213 

and  we  thus  complete  the  specification  of  the  single  force  to  which 
the  system  is  reduced  when  (22)  holds. 

579.  If  the  line  of  the  force  is  inclined  at  any  angle  to  the  plane 
of  the  couple,  the  resultant  system  can  be  further  reduced  by  §  569, 
to  a  smaller  couple  and  a  force  in  a  determinate  line,  the  'central 
axis.'  This  couple  is  GcosO,  and  according  to  the  notation,  may  be 
thus  expressed  by  §  464,  (7),  if  we  substitute  the  values  given  in  (19) 
and  (2o)> 

^        .     XL+YM+ZN  .     , 

Gcose^ (25) 

The  other  component  couple,  G  sin  B,  lies  in  the  same  plane  as  F, 
and  with  it  may  be  reduced  by  §568  to  one  force,  which  will  be 
parallel  to  F,  that  is,  in  the  direction  (/,  m,  «),  at  a  distance  from  it 

equal  to  — 5 —  .     Hence  the  direction  cosines  of  00'  will  be 

mv-rifi     n\-/v     /fi-mX  .  ,, 

sind    '   "ii^'  "imT"  ^^^ 

Substituting  ia  each  of  these  for  /,  X,  &c.,  their  respective  values, 

and  multiplying  each  member  by  — ^ —  ,  we  have  for  the  co-ordinates 

of  the  point  C^,  as  in  §  578, 

YN-ZM      ZL~XN      XM-  YL  . 

F''       '  F''       *  'F'       •  ^^^^ 

A  single  force,  F,  through  the  point  thus  specified  in  the  direction 
(/,  tn,  n),  with  a  couple  in  a  plane  perpendicular  to  it,  and  having 

XL  +  YM-^ZN 
F 

for  its  moment,  is  consequently  the  system  oi  force  along  central  axis 
and  minimum  couple,  to  which  the  given  set  of  forces  is  determinately 
reducible  by  Poinsot's  beautiful  method. 

580.  The  position  of  the  central  axis  may  be  determined  other- 
wise; tiius,  instead  of  in  the  first  place  bringing  the  forces  to  C,  bring 
them  to  any  point  7",  of  which  let  {x,  y^  z)  be  the  co-ordinates.  Then 
instead  of  l^z^+Y^z^  +  &c.,  which  we  had  before  (§  574),  we  have  now 

K(z,-z)+Y,{z,-z)-^kc, 

or  Y,  z,  +  Y,z,  +  &c.  -{Y,  +  Y,  +  &c.)  z, 

and  so  for  the  others.  Then  for  the  moments  of  the  couples  of  trans- 
ference we  have 

n  =  L  -{Zy-Yz), 
i«  =  M-  (Xz  -  Zx), 
B==N-{Yx-Xy). 
Now,  let  T  be  chosen,  if  possible,  so  as  to  make  the  resultant 


814  ABSTRACT  DYNAMICS. 

couple  lie  in  a  planQ  perpendicular  to  it.    The  condition  to  be  ful- 
filled in  this  case  is 

which,  when  for  %,  &c.,  we  substitute  their  values,  becomes, 

Z-(Zy-Yz)     M-^{Xz-Zx)     N-{Yx-Xy) 
_ = , 

which  is  the  equation  of  the  central  axis  of  the  system. 

To  show  that  O*,  the  point  determined  in  §§  578,  579,  is  in  the 
cehtral  axis  thus  found ;  we  have,  substituting  for  x,  y,  z,  the  values 
given  in  (24), 

-     Z(ZL-XJV)+Y(XM-YZ) 

^         '"    .?'  „ 

— — — —  =  &c. 


■Reducing,  and  remarking  that 

ZJi'-ZY'-ZZ'^ZX*,. 
we  find  that  the  first  member  becomes 

and  is  therefore  equal  to  each  of  the  two  others.    Thus  is  verified  the 
comparison  of  the  two  methods. 

581.  In  one  respect,  this  reduction  of  a  system  of  forces  to  a 
couple,  and  a  force  perpendicular  to  its  plane,  is  the  best  and  simplest, 
especially  in  having  the  advantage  of  being  determinate,  and  it  gives 
very  clear  and  useful  conceptions  regarding  the  effect  of  force  on 
a  rigid  body.  The  system  may,  however,  be  farther  reduced  to  two 
equal  forces  acting  symmetrically  on  the  rigid  body,  but  whose  po- 
sition is  indeterminate.  Thus,  supposing  the  central  axis  of  the 
system  has  been  found,  draw  a  line  AA',  at  right  angles  through  any 
point  C  in  it,  so  that  CA  may  be  equal  to  CA'.  For  J^,  acting  along 
the  central  axis,  substitute  |i?  at  each  end  of  AA'.  Thus,  choosing 
this  line  AA'  as  the  arm  of  the  couple,  and  calling  it  a,  we  have  at 

G 

each  extremity  of  it  two  forces,  —  perpendicular  to  the  central  axis, 

and  1^  parallel  to  the  central  axis.     Compounding  these,  we  get 

(G\h 
IJi^  +  -J-)   t    through   A    and   A'  re- 
spectively, perpendicular  to  AA',  and  equally  inclined  at  the  angle 
tan"'  — jj  on  the  two  sides  of  the  plane  through  AA'  and  the  central 
axis. 

582.  It  is  obvious,  from  the  formulae  of  §  195,  that  if  masses  pro- 
portional to  the  forces  be  placed  at  the  several  points  of  application 
of  these  forces,  the  centre  -of  inertia  of  these  masses  will  be  the  same 


STATICS   OF  SOLIDS  AND  FLUIDS.  215 

point  in  the  body  as  the  centre  of  parallel  forces.  Hence  the  re- 
actions of  the  different  parts  of  a  rigid  body  against  acceleration  in 
parallel  lines  are  rigorously  reducible  to  one  force,  acting  at  the  centre 
of  inertia.  The  same  is  true  approximately  of  the  action  of  gravity 
(m  a  rigid  body  of  small  dimensions  relatively  to  the  eartli,  and  hence 
the  centre  of  inertia  is  sometimes  (§  195)  called  the  Centre  of  Qrcvify. 
But,  except  on  a  centrobaric  body  (§  543),  gravity  is  not  in  general 
reducible  to  a  single  force  •  and  when  it  is  so,  this  force  does  not  pass 
through  a  point  fixed  relatively  to  the  body  in  all  positions. 

583.  The  resultant  of  a  system  of  parallel  forces  is  not  a  single 
force  j,vhen  the  algebraic  sum  of  the  given  forces  vanishes.  In 
this  case  the  resultant  is  a  couple  whose  plane  is  parallel  to  the 
common  direction  of  the  forces.  A  good  example  of  this  is  furnished 
by  a  magnetized  mass  of  steel,  of  moderate  dimensions,  subject  to  the 
influence  of  the  earth's  magnetism  only.  As  will  be  sho^Ti  later,  the 
amounts  of  the  so-called  north  and  south  magnetisms  in  each  element 
of  the  mass  are  equal,  and  are  therefore  subject  to  equal  and  opposite 
forces,  all  parallel  to  the  line  of  dip.  Thus  a  compass-needle  expe- 
riences from  the  earth's  magnetism  merely  a  couple  or  directive  action, 
and  is  not  attracted  or  repelled  as  a  whole. 

584.  If  three  forces,  acting  on  a  rigid  body,  produce  equilibrium, 
their  directions  must  lie  in  one  plane;  and  must  all  meet  in  one  point, 
or  be  parallel.  For  the  proof,  we  may  introduce  a  consideration 
which  will  be  very  useful  to  us  in  investigations  connected  with  the 
statics  of  flexible  bodies  and  fluids. 

If  any  forces,  acting  on  a  solid  or  fluid  body,  produce  equilibrium,  we 
may  suppose  any  portions  of  the  body  to  become  fixed,  or  rigid,  or  rigid 
and  fixed,  zcithout  destroying  the  equilibrium. 

Applying  this  principle  to  the  case  above,  suppose  any  two  points 
of  the  body,  respectively  in  the  lines  of  action  of^  two  of  the  forces,  to 
be  fixed — the  third  force  must  have  no  moment  along  the  line  joining 
these  points;  that  is,  its  direction  must  pass  through  the  line  joining 
them.  As  any  two  points  in  the  lines  of  action  may  be  taken,  it 
follows  that  the  three  forces  are  coplanar.  And  three  forces  in  one 
plane  cannot  equilibrate,  unless  their  directions  a.'-e  parallel  or  pass 
through  a  point, 

585.  It  is  easy  and  useful  to  consider  various  cases  of  equilibrium 
when  no  forces  act  on  a  rigid  body  but  gravity  and  the  pressures, 
normal  or  tangential,  between  it  and  fixed  supports.  Thus,  if  one 
given  point  only  of  the  body  be  fixed,  it  is  evident  that  the  centre  of 
gravity  must  be  in  the  vertical  line  through  this  point — else  the  weiglit 
and  the  reaction  of  the  support  would  form  an  unbalanced  couple. 
Also  for  stable  equilibrium  the  centre  of  gravity  must  be  belou-i  the 
point  of  suspension.  Thus  a  body  of  any  form  may  be  made  to 
stand  in  stable  equilibrium  on  the  point  of  a  needle  if  we  rigidly 
attach  to  it  such  a  mass  as  to  cause  the  joint  centre  of  gravity  to  be 
below  the  point  of  the  needle. 


2l6 


ABSTRACT  DYNAMICS. 


586.  An  interesting  case  of  equilibrium  is  suggested  by  what  are 
called  Rocking  Stones,  where,  whether  by  natural  or. by  artificial  pro- 
cesses, the  lower  surface  of  a  loose  mass  of  rock  is  worn  into  a  convex 
form  which  may  be  approximately  spherical,  while  the  bed  of  rock  on 
which  it  rests  m  equilibrium  is,  whether  convex  or  concave,  also  ap- 
proximately spherical,  if  not  plane.  A  loaded  sphere  resting  on  a 
spherical  surface  is  therefore  a  type  of  such  cases. 

Let  O,  0'  be  the  centres  of  curvature  of  the  fixed  and  rocking 
bodies  respectively,  when  in  the  position  of  equilibrium. 
Take  any  two  infinitely  small  equal  arcs  PQ^  Fp ;  and 
at  Q  make  the  angle  O' QR  equal  to  FOp.  When,  by 
displacement,  Q  and  /  become  the  points  in  contact, 
QR  will  evidently  be  vertical ;  and,  if  the  centre  of 
gravity  G,  which  must  be  in  OFO"  when  the  movable 
body  is  in  its  position  of  equilibrium,  be  to  the  left  of 
QR,  the  equilibrium  will  obviously  be  stable.  Hence, 
if  it  be  below  R,  the  equilibrium  is  stable,  and  not 
unless. 

Now  if  p  and  cr  be  the  radii  of  curvature  OF,  OF 
of  the  two  surfaces,  and  6  the  angle  FOp,  the  angle 

QO'R  will  be  equal  to  —  j  and  we  have  in  the  triangle 


QOF{ 


RO 


sm 


Hence  PR  =  <t 

(7  +  p 

and  therefore,  for  stable  equilibrium, 


: :  cr ;  (7  +  p  (approximately). 

p  +  cr' 


FG< 


pq- 

p  +<r' 


If  the  lower  surface  be  plane,  p  is  infinite,  and  the  condition  becomes 
(as  in  §  256) 

FG<a: 
If  the  lower  surface  be  concave,  the  sign  of  p  must  be  changed,  and 
the  condition  becomes 


FG< 


per 

p-(7' 


which  cannot  be  negative,  since  p  musi  be  numericaHy  greater  than  <r 
in  this  case. 

587.  If  two  points  hd  fixed,  the  only  motion  of  which  the  system  is 
capable  is  one  of  rotation  about  a  fixed  axis.  The  centre  of  gravity 
must  then  be  in  the  vertical  plane  passing  through  those  points,  and 
de/cnv  the  line  adjoining  them  for  stable  equilibrium. 

588.  If  a  rigid  body  rest  on  a  fixed  surface,  there  will  in  general  bo 
only  //iree  points  of  contact,  §  380 ;  and  the  body  will  be  in  stable 


STATICS  OF  SOLIDS  AXD   FLUIDS  2ij 

equilibrium  if  the  vortical  line  drawn  from  its  centre  of  gravity  cuts 
the  plane  of  these  three  pomts  uuithin  the  triangle  of  which  they  form 
the  corners.  For  if  one  of  these  supports  be  removed,  the  body  will 
obviouily  tend  to  fall  towards  that  support.  Hence  each  of  the  three 
prevents  the  body  from  rotating  about  the  line  joining  the  other -two. 
Thus,  for  instance,  a  body  stands  stably  on  an  inclined  plane  (if  the 
friction  be  sufficient  to  prevent  it  from  sliding  down)  when  the  vertical 
line  drawn  through  its  centre  of  gravity  falls  within  the  base,  or  area 
bounded  by  the  shortest  line  which  can  be  drawn  round  the  portion  in 
contact  with  the  plane.  Hence  a  body,  which  cannot  stand  on  a 
horizontal  plane,  may  stand  on  an  inclined  plane. 

5S9.  A  curious  theorem,  due  to  Pappus,  but  commonly  attributed 
to  Guldinus,  may  be  mentioned  here,  as  it  is  employed  with  advantage 
in  some  cases  in  finding  the  centre  of  gravity  of  a  body — though  it  is 
really  one  of  the  geometrical  properties  of  the  Centre  of  Inertia.  It  is 
obvious  from  §  195.  If  a  plane  dosed  curve  rrcolve  through  any  an^^Ie 
about  an  axis  in  its  plane,  the  solid  content  of  the  surfaa  generated  is 
equal  to  the  product  of  the  area  of  e  it  fur  end  into  t/ie  length  of  the  path 
described  by  its  centre  of  gravity ;  and  the  area  of  the  curved  surface  is 
equal  to  the  product  of  the  length  of  the  curve  into  tJu  le?igth  of  t/ic  path 
described  by  its  centre  of  gravity. 

590.  The  general  principles  upon  which  forces  of  constraint  and 
friction  are  to  be  treated  have  been  stated  above  (§3  258,  405).  We 
add  here  a  few  examples,  for  the  sake  of  illustrating  the  applicltion 
of  these  principles  to  the  equilibrium  of  a  rigid  body  in  some  of  the 
more  important  practical  cases  of  constraint. 

591.  The  application  of  statical  principles  to  the  Mec/ianical  Po-wers^ 
or  elementar)'  machines,  and  to  their  combinations,  however  complex, 
requires  merely  a  statement  of  their  kmematical  relations  (as  in  §^  91, 
97,  113,  &:c.)  and  an  immediate  translation  into  Dynamics  by  New- 
ton's principle  (,§  241);  or  by  Lagrange's  Virtual  Velocities  (§  254), 
with  special  attention  to  the  introduction  of  forces  of  friction,  as  m 
§  405.  In  no  case  can  this  process  involve  further  difficulties  than 
are  implied  in  seeking  the  geometrical  circumstances  of  any  infinitely 
small  disturbance,  and  in  the  subsequent  solution  of  the  equations  to 
which  the  translation  into  dynamics  leads  us.  We  will  not,  therefore, 
stop  to  discuss  any  of  these  questions;  but  will  take  a  few  examples 
of  no  very  great  difficulty,  before  for  a  time  quitting  this  part  of  the 
subject.  The  principles  already  developed  will  be  of  constant  use  to 
us  in  the  remainder  of  the  work,  which  will  furnish  us  with  ever- 
recurring  opportunities  of  exemplifying  their  use  and  mode  of  appli- 
cation. 

Let  us  begin  with  the  case  of  the  Balance,  of  which  we  promised  I 
(§  384)  to  give  an  investigation. 

592.  Ex.  I.  We  will  assume  the  line  joining  the  points  of  attach- 
ment of  the  scale-pans  to  the  arms  to  be  at  right  angles  to  the  line 
joining  the  centre  of  gravity  of  the  beam  with  the  fulcrum.     It  is 


2 1 8  ABSTRA  CT  D  YNAMIC^. 

obvious  that  the  centre  of  gravity  of  the  beam  must  not  comcide  with 
the  knife-edge,  else  the  beam  would  rest  mdifferently  m  any  position. 
We  will  suppose,  in  the  first  place,  that  the  arms  are  not  of  equal  length. 

D  Let  O  be  the  fulcrum,  G  the 

centre  of  gravity  of  the  beam, 
^  M  its  mass ;  and  suppose  that 
with  loads  P  and  Q  in  the  pans 
the  beam  rests  (as  drawn)  in  a 
position  making  an  angle  Q  with 
the  horizontal  line. 

Taking    moments    about    0, 
and,  for  convenience  (see  §  185), 
using  gravitation  measurement  of  the  forces,  we  have 

<2 {AB cos  e^OA  sin  Q)  +  M.  OG slnO ^ B (AC cos  $  -  OA  sin  6). 

From  this  we  find 

.      ^_      P.AC-Q.AB 

{B+  Q)OA  +  M.OG' 
If  the  arms  be  equal  we  have 

tang-  (^-Q)^^ 

{B+Q)OA  +  M.OG' 

Hence  the  Sensibility  (§  384)  is  greater,  (i)  as  the  arms  are  longer, 
(2)  as  the  mass  of  the  beam  is  less,  (3)  as  the  fulcrum  is  nearer  to  the 
line  joining  the  points  of  attachment  of  the  pans,  (4)  as  the  fulcrum  is 
nearer  to  the  centre  of  gravity  of  the  beam.  If  the  fulcrum  be  m  the 
line  joining  the  points  of  attachment  of  the  pans,  the  sensibility  is  the 
same  for  the  same  difference  of  loads  in  the  pan. 

To  determine  the  Stability  we  must  investigate  the  time  of  oscilla- 
tion of  the  balance  when  slightly  disturbed.  It  will  be  seen,  by  refer- 
ence to  a  future  chapter,  that  the  equation  of  motion  is  approximately 

\Mk'  ^{P+Q)  0B'\  6  +  Qg{AB  cos  6  +  OA  sin  6) 

+  MgOG  sin  e-Pg {AC cos  6-  OA  sin  6)  =  o, 

k  being  the  radius  of  gyration  (§"235)  of  the  beam.  If  we  suppose 
the  arms  and  their  loads  equal,  we  have  for  the  time  of  an  infinitely 
small  oscillation' 

TT        /       Mk'+2P.OB' 

V  {2P.0A+M.0G)g' 

Thus  the  stability  is  greater  for  a  given  load,  (i)  the  less  the  length  of 
the  beam,  (2)  the  less  its  mass,  (3)  the  less  its  radius  of  g>Tation,  (4) 
the  further  the  fulcrum  from  the  beam,  and  from  its  centre  of  gravity. 
With  the  exception  of  the  second,  these  adjustments  are  the  very 
opposite  of  those  required  for  sensibihty.  Hence  all  we  can  do  is  to 
effect  a  judicious  compromise;  but  the  less  the  mass  of  the  beam,  the 
better  will  the  balance  be,  in  boih  respects. 

The  general  equation,  above  v,Titten,  shows  that  if  the  length,  and* 
the  radius  of  g>Tation,  of  one  arm  be  diminished,  the  corresponding. 


STATICS   OF  SOLIDS   AND   FLUIDS. 


2ig 


load  being  increased  so  as  to  maintain  equilibrium — a  form  of  balance 
occasionally  useful — the  sensibility  is  increased. 

Ex.  II.  Find  the  position  of  equilibrium  of  a  rod  AB  resting  on  a 
smooth  horizontal  rail  D,  its  lower  end  pressing  against  a  smooth 
vertical  wall  AC  parallel  to  the  rail. 

The  figure  represents  a  vertical  section  through  tht?  rod,  which 
must  evidently  be  in  a  plane  perpendicular  to  the  wall  and  rail. 

The  only  forces  acting  are  three,  R  the  pressure  of  the  wall  on  the 
rod,  horizontal;  .5  that  of  the 
rail  on  the  rod,  perpendicular 
to  the  rod;  W  the  weight  of 
the  rod,  acting  vertically  down- 
wards at  its  centre  of  gravity. 
If  the  half-length  of  the  rod  be 
a,  and  the  distance  of  the  rail 
from  the  wall  b,  these  are  given 
— and  all  that  is  wanted  to  ftx 
the  position  of  equilibrium  is  .the. angle  the  rod  makes  with. the  wall. 


sin  6 


Call  CAB,  b.     Then  we  see  at  once  that  AD  ^ 

Resolving  horizontally         R-  S  cos6  ~o,  (i) 

vertically  IV-  S^'wiB  -o.  (2) 

Taking  moments  about  A, 

S.AD-  Jl'a  sin  y=o, 

or  Sl>  -  JVasin'd  =  0.  (3) 

As  there  are  only  three  unknown  quantities,  R,  S,  and  9,  these  three 
equations  contain  the  complete  solution  of  the  problem.  By  (?) 
and  (3) 

sin'^  =  -■,  which  gives  9. 


IV 
Hence  by  (2)  5=-^—, 

^  ^  '  Sin  9 

and  by  (i)  R  =  Scos9  =  lVcoi9. 

Ex.  III.  As  an  additional  example,  suppose  the  wall  and  rail  to  be 
rough,  and  /x  to  be  the  co-efficient  of  statical  friction  for  both.  If  the 
rod  be  placed  in  the  position  of  equilibrium  just  investigated  for  the 
case  of  no  friction,  none  will  be  called  into  play,  for  there  will  be  no 
tendency  to  motion  to  be  overcome.  If  the  end  A  be  brought  lowei 
and  lower,  more  and  more  friction  will  be  called  into  play  to  over- 
come the  tendency  of  the  rod  to  fall  between  the  wall  and  the  rail, 
until  we  come  to  a  limiting  position  in  which  motion  is  about  to 
commence.     At  that  instant  the  friction  at  /i  is  /i  times  the  pressure- 


320 


ABSTRACT  DYNAMICS. 


on  the  wall,  and  acts  upwards.  That  at  i?  is  /x  times  the  pressure 
on  the  rod,  and  acts  in  the  direction  DB,  Calling  CAD  =  Q^  in  this 
case,  our  three  equations  become 

J?i  -»•  M'S',  sin  ^1  -  ^  cos  ^j  =  o,  (i  ^) 

W^-/iJ?,-6isin^i-/x^jC0S^i=o,  (zj 

S^b-Wa%\Xi^Q^  =0.  (3i) 

The  directions  of  both  the  friction-forces  passing  through  A,  neither 
appears  in  (jj.    This  is  why  A  is  preferable  to  any  other  point  about 
which  to  take  moments. 
By  eliminating  R^  and  S-^  from  these  equations  we  get 

1  -  -7  sm'  ^i  =  /^  T  sin^  ^i  (2  cos  ^,  -  m  sin  B^,  (4,) 

from  which  ^,  is  to  be  found.  Then  S^  is  known  from  (3  J,  and  R^ 
from  either  of  the  others. 

If  the  end  A  be  raised 'above  the  position  of  equilibrium  without 
friction,  the  tendency  is  for  the  rod  to  fall  outside  the  rail;  more  and 
more  friction  will  be  called  into  play,  till  the  position  of  the  rod  {d^ 
is  such  that  the  friction  reaches  its  greatest  value,  /x  times  the  pressure. 
We  may  thus  find  another  limiting  position  for  stability;  and  between 
these  the  rod  is  in  equilibrium  in  any  position. 

It  is  useful  to  observe  that  in  this  second  case  the  direction  of  each 
friction  is  the  opposite  to  that  in  the  former,  and  the  same  equations 
will  serve  for  both  if  we  adopt  the  analytical  artifice  of  changing  the 
sign  of  ft.     Thus  for  ^n,  by  (41), 


a  .  , 
I  -  T  sm" 

t> 


^s  =  -  M  T  sin^^a  (2  cos  ^2  +  /x  sin  B^. 


(4.) 


B^x.  IV.  A  rectangular  block  lies  on  a  rough  horizontal  plane,  and 
is  acted  on  by  a  horizontal  force  whose  line  of  action  is  midway  be- 
tween two  of  the  ver- 
tical sides.  Find  the 
magnitude  of  the  force 
when  just  sufficient  to 
produce  motion,  and 
whether  the  motion  will 
be  of  the  nature  of  slid- 
ing  or  cmcrtvrning. 

If  the  force  P  tends  to 
overturn  the  body,  it  is 
'evident  that  it  will  turn  about  the  edge  .i^.and  therefore  the  pressure,  R, 
of  the  plane  and  the  friction,  5.  act  at  that  edge.  Our  statical  condi- 
tions are,  of  course  R-  W 

S=B 

where  b"\s  half  the  length  of  the  solid,  and  a  the  distance  of  /'from 
the  plane.     From  these  we  have  S=  -  W. 


STATICS  OF  SOLIDS  AND  FLUIDS. 


321 


Now  S  cannot  exceed  "j^,  whence  we  must  not  have  -  greater 

than  ^,  if  it  is  to  be  possible  to  upset  the  body  by  a  horizontal  force 
in  the  line  given  for  P 

A  simple  geometrical  construction  enables  us  to  solve  this  andsirmlar 
problems,  and  will  be  seen  at  once  to  be  merely  a  graphic  representa- 
tion o{  the  above  process.  Thus  if  we  produce  the  directions  of 
the  apphed  force,  and  of  the  weight,  to  meet  in  H,  and  make  at  A  the 
angle  BAK  whose  co-tangent  is  the  co-efficient  of  friction :  there  will 
be  a  tendenc  to  upset,  or  not,  according  as  ^is  above,  or  below,  AK. 

Ex,  V.  A  mass  such  as  a  gate,  is  supported  by  two  rings,  A  and 
B,  which  pass  looseiy  round  a 
smooth  vertical  post  In  equi- 
librium, it  is  obvious  that  at  A 
the  part  of  the  ring  nearest  the 
mass,  and  at  B  the  farthest  Trom 
it,  will  be  in  contact  with  the 
post.  The  pressures  exerted  on 
the  rings,  R  and  5.  will  evi- 
dently have  the  directions  AC^ 
CB,  indicated  in  the  diagram.  If  no  other  force  besides  gravity  act 
on  the  mass,  the  Ime  of  action  of  its  weight,  W,  must  pass  through 
the  point  C  (§  584).  And  it  is  obvious  that,  however  small  be  the 
co-efficient  of  friction,  provided  there  be  friction  at  all,  equilibrium 
is  always  possible  if  the  distance  of  the  centre  of  gravity  from  the 
post  be  great  enough  compared  with  the  distance  between  the 
rings. 

When  the  mass  is  just  about  to  slide  down,  the  full  amount  of 
friction  is  called  into  play,  and  the  angles  which  R  and  .S  make  with 
the  horizon  are  each  equal  to  the  sliding  angle.  If  we  draw  AC, 
BC  according  to  this  condition,  then  for  equilibrium  the  centre  of 
gravity  G  must  not  lie  between  the  post  and  the  vertical  line  through 
the  point  Cthus  determined.  If,  as  in  the  figure,  G  lies  in  the  ver- 
tical Une  through  C,  then  a  force  applied  upwards  at  Q^,  or  down- 
wards at  Q,,  will  remove  the  tendency  to  fall;  but  a  force  applied 
upwards  at  Q^,  or  ddwnwards  at  (2^,  will  produce  sliding  at  once. 

A  similar  investigation  is  easily  applied  to  the  jamming  of  a  sliding 
piece  or  drawer,  and  to  the  determination  of  the  proper  point  of  appU-' 
cation  of  a  force  to  move  it.     This  we  leave  to  the  student. 

As  an  illustration  of  the  use  of  friction,  let  us  consider  a  ccrd 
wound  round  a  rough  cylinder,  and  on  the  point  of  sliding. 

Neglecting  the  weight  of  the  cord,  which  is  small  in  practice  com- 
pared with  the  other  forces;  and  con- 
sidering a    small    portion  AB   of  the  ^^ 
cord,    such    that   the   tangents   at    its 
extremities  include  a  very  small  angle   - — -''^^  JJ 
0;   let  y  be  the  tension  at  one  end, 


222  ABSTRACT  DYNAMICS. 

T'  at  the  other,  /  the  pressure  of  the  rope  on  the  'cylinder  per  unit 
of  length. 

Q 

Then/>.AB  =  2  7"  sin  -  =  TO  approximately.    Also  ii.p.AB=T  -T 

when  the  rope  is  just  about  to  slip,  i.e. 

y^TQ^r-T, 
or  r={i+y.6)T. 

Hence,  for  equal  small  deflections,  6,  of  the  rope,  the  tension 
increases  in  the  geometrical  ratio  (1  +  1x6) :  i;  and  thus  by  a  common 
theorem  (compound  interest  payable  every  instant)  we  have  T=  i^'^T^, 
if  7",  T^,  be  the  tensions  at  the  ends  of  a  cord  wrapped  on  a  cylinder, 
when  the  external  angle  between  the  directions  of  the  free  ends  is  a. 
[<  is  the  base  of  Napier's  Logarithms.]  We  thus  obtain  the  singular 
result,  that  the  dimensions  of  the  cylinder  have  no  influence  on  the 
increase  of  tension  by  friction,  provided  the  cord  is  perfectly  flexible. 

593.  Having  thus  briefly  considered  the  equilibrium  of  a  rigid 
body,  we  propose,  before  entering  upon  the  subject  of  deformation 

/©f  elastic  solids,  to  consider  certain  intermediate  cases,  in  each  of 
which  a  particular  assumption  is  .made  the  basis  of  the  investiga- 
tion— thereby  avoiding .  k  very  considerable  amount  of  analytical 
difficulties. 

594.  Very  exceHent  examples  of  this  kind  are  furnished  by  the 
statics  of  a  flexible  and  fhextensible  cord  or  chain,  fixed  at  both  ends, 
and  subject  to  the  action  of  any  forces.  The  curve  in  which  the 
chain  hangs  in  any  case  may '-be  called  a  Catenary,  although  the  term 
is  usually  restricted  to  the  case  of  a  uniform  chain  acted  on  by  gravity 
only. 

595.  We  may  consider  separately  the  conditions  of  equilibrium  of 
each  element;  or  we  may  apply  the  general  condition  (§  257)  that  the 
whole  potential  energy  is  a  minimum,  in  the  case  of  any  conservative 
.system  of  forces;  or,  especially  when  gravity  is  the  only  external 
force,  we  may  consider  the  equilibrium  of  2i  finite  portion  of  the  chain 
treated  for  the  time  as  a  rigid  body  (§  584). 

596.  The  first  of  these  methods  gives  immediately  the  three  follow- 
ing equations  of, equilibrium,  for  the  catenary  in  general: — 

(i)  The  rate  of  variation  of  the  tension  per  unit  of  length  along 
the  cord  is  equal  to  the  tangential  component  of  the  applied  force, 
per  ynit.  of  length. 

(2)  The  plane  oiP  Curvature  of  the  cord  contains  the  normal  com- 
ponent of  the  applied  force'  and  the  centre  of  curvature  is  on  the 
opposite  side  of  the  arc  from  that  towards  which  this  force  acts. 

(3)  The  amount  of  the  curvature  is  equal  to  the  normal  componeni 
of  the  applied  force  per  unit  of  length  at  any  point  divided  by  the  ten- 
sion of  the  cord  at  the  same  point. 

The  first  of  these  is  simply  the  equation  of  equilibrium  of  an 
infinitely  small  element  of  the  cord  relatively  to  tangential  motion. 
The  second  and  third  express  that  the  component  of  the  resultant 


STATICS  OF  SOLIDS  AND  FLUIDS.  223 

of  the  tensions  at  the  two  ends  of  an  infinitely  small  arc,  along  tfie 
normal  through  its  middle  point,  is  directly  opposed  and  is  equal  to 
the  normal  applied  force,  and  is  equal  to  the  whole  amount  of  it  or» 
the  arc  For  the  plane  of  the  tangent  lines  in  which  those  tensions 
act  is  (S  12)  the  plane  of  curvature.  And  if  ^  be  the  angle  between 
them  (or  the  infinitely  small  angle  by  which  the  angle  between  their 
positive  directions  falls  short  of  ir),  and  T  the  arithmetical  mean  of 
their  magnitudes,  the  component  of  their  resultant  along  the  line 
bisecting  the  angle  between  their  positive  directions  is  2Z'sin  \9,  rigor- 
ously: or  TO,  smce  6  is  infinitely  small  Hence  7T9  =  iV5j  if  hs  be 
the  length  of  the  arc,  and  N^s  the  whole  amount  of  normal  force 

applied  to  it.    But  (§  9)  ^  =  —  if  p  be  the  radius  of  curvature;  and 

therefore  —=-:=,, 

P      -i 

which  is  the  equation  stated  in  words  (3)  above. 

597.  From  (i)  of  §  596,  we  see  that  if  the  applied  forces  on  any 
particle  of  the  cord  constitute  a  conservative  system,  and  if  any  equal 
infinitely  small  lengths  of  the  string  experience  the  same  force  and 
in  the  same  direction  when  brought  into  any  one  position  by  motion 
of  the  string,  the  difl'erence  of  the  tensions  of  the  cord  at  any  two 
points  of  it  when  hanging  in  equilibrium,  is  equal  to  the  difference 
of  the  potential  (§  504)  of  the  forces  between  the  positions  occupied 
by  these  points.  Hence,  whatever  the  position  where  the  potential  is 
reckoned  zero,  the  tension  of  the  string  at  any  point  is  equal  to  the 
potential  at  the  position  occupied  by  it,  with  a  constant  added. 

598.  From  §  596  it  follows  immediately  that  if  a  material  particle 
of  unit  mass  be  carried  along  any  catenary  with  a  velocity,  s,  equal 
to  T,  the  numerical  measure  of  the  tension  at  any  point,  the  force 
upon  it  by  which  this  is  done  is  in  the  same  direction  as  the  resultant 
of  the  apphed  force  on  the  catenary  at  this  point,  and  is  equal  to 
the  amount  of  this  force  per  unit  of  length,  multiplied  by  T.  For 
denoting  by  .S  the  tangential,  and  (as  before)  by  N  the  normal 
component  of  the  applied  force  per  unit  of  length  at  any  point  J^ 
of  the  catenary,  we  have,  by  §  596  (i),  5  for  the  rate  of  variation  oi" 
s  per  unit  length,  and  therefore  Si  for  its  variation  per  unit  of  time. 

That  is  to  say,  s  =  Ss  =  ST, 

or  (§  225)  the  tangential  componerit  force  on  the  moving  particle 
is  equal  to  ST.     Again,  by  §  596  (3), 

NT=  -  =  -  , 
P       P 

or  the  centrifugal  force  of  the  moving  particle  in  the  circle  of  cur- 
vature of  its  path,  that  is  to  say,  the  normal  component  of  the 
force  on  it,  is  equal  to  NT.  And  lastly,  by  (2)  this  force  is  io 
the  same  direction  as  N.     We  see  tb.sreforc  that  the  direction  of  the 


Lor      /^. 

2fz  V     23 


224  ABSTRACT  DYNAMICS. 

whole  force  on  the  moving  particle  is  the  same-  as  that  of  the 
resultant  of  5  and  N\  and  its  magnitude  is  T  times  the  magnitude 
of  this  resultant 

599.  Thus  we  see  how,  from  the  more  familiar  problems  of  the 
kinetics  of  a  particle,  we  may  immediately  derive  curious  cases 
of  catenaries.  Fi>r  instance :  a  particle  under  the  influence  of  a 
constant  force  in  parallel  lines  moves  in  a  parabola  with  its  axis 
vertical,  with  velocity  at  each  point  equal  to  that  generated  by 
the  force  acting  through  a  space  equal  to  its  distance  from  the 
directrix.  Hence,  if  z  denote  this_distance,  and /the  constant 
force,  T=  Jt/z 

in  the  allied  parabolic  catenary;  and  the  force  on  the  catenary  is 
parallel  to  the  axis,  and  is  equal  in  amount  per  unit  of  length,  to 

J2fz 

Hence  if  the  force  on  the  catenary  be  that  of  gravity,  it  must  have 
its  axis  vertical  (its  vertex  downwards  of  course  for  stable  equili- 
brium) and  its  mass  per  unit  length  at  any  point  must  be  inversely 
as  the  square  root  of  the  distance  of  this  point  above  the  directrix. 
From  this  it  follows  that  the  whole  weight  of  any  arc  of  it  is 
proportional  to  its  horizontal  projection. 

600.  Or,  if  the  question  be,  to  find  what  force  towards  a  given 
fixed  point,  will  cause  a  cord  to  hang  in  any  given  plane  curve  with 
this  point  in  its  plane;  it  may  be  answered  immediately  from  the 
solution  of  the  coresponding  problem  in  'central  forces.' 

601.  When  a  perfectly  flexible  string  is  stretched  over  a  smooth 
surface,  and  acted  on  by  no  other  force  throughout  its  length  than 
the  resistance  of  this  surface,  it  will,  when  in  stable  equilibrium, 
lie  along  a  line  of  minimum  length  on  the  surface,  between  any 
two  of  its  points.  For  (§  584)  its  equilibrium  can  be  neither 
disturbed  nor  rendered  unstable  by  placing  staples  over  it,  through 
which  it  is  free  to  slip,  at  any  two  points  where  it  rests  on  the 
surface  and  for  the  intermediate  part  the  energy  criterion  of  stable 
equilibrium  is  that  just  stated. 

There  being  no  tangential  force  on  the  string  in  this  case,  and  the 
normal  force  upon  it  being  along  the  normal  to  the  surface,  its  oscu- 
lating plane  (§  596)  must  cut  the  surface  everywhere  at  right  angles. 
These  considerations,  easily  translated  into  pure  geometry,  establish 
the  fundamental  property  of  the  geodttic  lines  on  any  surface.  The 
analytical  investigations  of  the  question,  when  adapted  to  the  case 
of  a  chain  of  not  given  length,  stretched  between  two  given  points  on 
a  given  smooth  surface,  constitute  the  direct  analytical  demonstration 
of  this  property. 

In  this  case  it  is  obvious  that  the  tension  of  the  string  is  the  same 
at  every  point,  and  the  pressure  of  the  surface  upon  it  is  [§  596  (3)] 
at  each  point  proportional  to  the  curvature  of  the  string. 


STATICS   OF  SOLIDS  AND   FLUIDS.  225 

602.  No  real  surface  being  perfectly  smooth,  a  cord  or  chain  may 
rest  upon  it  when  stretched  over  so  great  a  length  of  a  geodetic  on  a 
convex  rigid  body  as  to  be  not  of  minimum  length  between  its 
extreme  points  :  but  practically,  as  m  tying  a  cord  round  a  ball, 
for  permanent  security  it  is  necessary,  by  staples  or  otherwise,  to 
constrain  it  from  lateral  slippmg  at  successive  points  near  enough 
to  one  another  to  make  each  free  portion  a  true  mmimum  on  the 
surface. 

603.  A  very  important  practical  case 'is  supplied  by  the  con- 
sideration of  a  rope  wound  round  a  rough  cylinder.  We  may 
suppose  it  to  lie  in  a  plane  perpendicular  to  the  axis,  as  we  thus 
simplify  the  question  very  considerably  without  sensibly  injuring 
the  utility  of  the  solution.  To  simplify  still  further,  we  shall  suppose 
that  no  forces  act  on  the  rope  but  tensions  and  the  reaction  of 
the  cylinder.  In  practice  this  is  equivalent  to  the  supposition  that 
the  tensions  and  reactions  are  very  large  compared  with  the  weight 
of  the  rope  or  chain ;  "which,  however,  is  inadmissible  in  some 
important  cases,  especially  such  as  occur  in  the  application  of  the 
principle  to  brakes  for  laying  submarine  cables,  to  ergometers,  and 
to  windlasses  (or  capstans  with  horizontal  axes). 

By  §  592  we  have  Z"-  T^f.^^, 

showing  that,  for  equal  successive  amounts  of  integral  curvature 
(§  14),  the  tension  of  the  rope  augments  in  geometrical  progression. 
To  give  an  idea  of  the  magnitudes  involved,  suppose  /x  =  -5,  ^  =  tt,  then 

T=T^c^^4.^iT^  roughly. 
Hence  if  the  rope  be  wound  three  times  round  the  post  or  cylinder 
the  ratio  of  the  tensions  of  its  ends,  when  motion  is  about  to  com- 
mence, IS 

(4.  81)''  :  I  or  about  12393  :  i. 

Thus  we  see  how,  by  the  aid  of  friction,  one  man  may  easily  check 
the  motion  of  the  largest  vessel,  by  the  simple  expedient  of  coiling  a 
rope  a  few  times  round  a  post.  This  application  of  friction  is 
of  great  importar^e  in  many  other  applications,  especially  to  ergome- 
ters  (sj  389,    39c   . 

604.  With  the  aid  of  the  preceding  investigations,  the  student 
may  easily  work  out  5or  himself  the  solution  of  the  general  problem 
of  a  cord  under  the  action  of  any  forces,  and  constrained  by  a 
rou^h  surface;  it  is  not  of  sufficient  importance  or  interest  to  find 
a  place  here. 

605.  An  elongated  body  of  elastic  material,  which  for  brevity 
we  shall  generally  call  a  wire,  bent  or  twisted  to  any  degree,  sub- 
ject only  to  the  condition  that  the  radius  of  curvature  and  the  reci- 
procal of  the  twist  are  everywhere  very  great  in  comparison  with 
the  greatest  transverse  dimension,  presents  a  case  in  which,  as  we 


2  26  ABSTRACT  DYNAMICS. 

shall  see,  the  solution  of  the  general  equations  for  the  equilibriurn 
of  an  elastic  solid  is  either  obtainable  m  finite  terms,  or  is  reducible 
to  comparatively  easy  questions  agreemg  m  mathematical  conditions 
with  some  of  the  most  elementary  problems  of  hydrokinetics,  elec- 
tricity, and  thermal  conduction.  And  it  is  only  for  the  determination 
of  certain  constants  depending  on  the  section  of  the  wire  and  the 
elastic  quality  of  its  substance,  which  measure  its  flexurai  and 
torsional  rigidity,  that  the  solutions  of  these  problems  are  required. 
When  th,e  corfstants  of  flexure  and  torsion  are  known,  as  we  shall 
now  suppose  them  to  be,  whether  from  theoretical  calculation  or 
experiment,  the  investigation  of  the  form  and  twist  of  any  length 
of  the  wire,  under  the  influence  of  any  forces  which  do  not  produce 
a  violation  of  the  condition  stated  above,  becomes  a  subject  of 
mathematical  analysis  involving  only  such  principles  and  formulae 
as  those  that  constitute  the  theory  of  curvature  (§§  9-15)  and  twist 
in  geometry  or  kinematics. 

606.  Before  entering  on  the  general  theory  of  elastic  solids,  we 
shal]  therefore,  according  to  the  plan  proposed  in  §  593,  examine 
the  dynamic  properties  and  investigate  the  conditions  of  equilibrium 
of  a  pertectly  elastic  wire,  without  admitting  any  other  condition  or 
ijfii'tation  of  the  circumstances  than  what  is  stated  in  §  605,  and 
without  assuming  any  special  quality  of  isotropy,  or  of  crystalline, 
fibrous  or  laminated  structure  in  the  substance. 

607.  Besides  showing  how  the  constants  of  flexurai  and  tor- 
sional rigidity  are  to  be  determined  theoretically  from  the  form  of 
the  transverse  secdon  of  the  wire,  and  the  proper  data  as  to  the 
elastic  qualities  of  its  substance,  the  complete  theory  simply  in- 
dicates that,  provided  the  conditional  limit  of  deformation  is  not 
exceeded,  the  following  laws  will  be  obeyed  by  the  wire. under 
stress : — 

Let  the  whole  mutual  action  between -the  parts  of  the.  wire  on  the 
two  sides  of  the  cross  section  at  any  point  (being  of  course  the  action 
of  the  matter  infinitely  near  this  plane  on  one  side,  upon  the  matter 
infinitely  near  it  on  the  other  side),  be  reduced  to  a  single  force 
through  any  point  of  the  section  and  a  single  couple.     Then — 

I.  The  twist  and  curvature  of  the  wire  in  the  neighbourhood  of 
this  section  are  independent  of  the  force,  and  depend  solely  on  the 
couple. 

II.  The  curvatures  and  rates  of  twist  producible  by  any  several 
couples  separately,  constitute,  if  geometrically  compounded,  the  curva- 
ture and  rate  of  twist  which  are  actually  produced  by  a  mutual  action 
equal  to  the  resultant  of  those  couples. 

608.  Jt  may  be  added,  although  not  necessary  for  our  present 
purpose,  that  there  is  one  determinate  point  in  the  cross  section 
such  that  if  it  be  chosen  as  the  point  to  which  the  forces  are  trans- 
ferred, a  higher  order  of  approximation  is  obtained  for  the  fultilraent 


STATICS  OF  SOLIDS  AND  FLUIDS.  227 

of  these  laws  than  if  any  other  point  of  the  section  be  taken-  That 
point,  which  in  the  case  of  a  wire  of  substance  uniform  through' its 
cross  section  is  the  centre  of  inertia  of  the  area  of  the  section,  we 
shall  generally  call  the  elastic  centre,  or  the  centre  of  elasticity,  of 
the  section.  It  has  also  the  following  important  property  : — The  line 
of  elastic  centres,  or,  as  we  shall  call  it,  the  elastic  central  line,  remains 
sensibly  unchanged  in  length  to  whatever  stress  within  our  conditional 
limits  (§  605)  the  wire  be  subjected.  The  elongation  or  contraction 
produced  by  the  neglected  resultant  force,  if  this  is  in  such  a  direction 
as  to  produce  any,  will  cause  the  line  of  rigorously  no  elongation  to 
deviate  only  infinitesimally  from  the  elastic  central  line,  m  any  part 
of  the  wire  finitely  curved.  It  will,  however,  clearly  cause  there  to 
be  no  line  of  rigorously  unchanged  length,  in  any  straight  part  of  the 
wire :  but  as  the  whole  elongation  would  be  infinitesimal  in  com- 
parison with  the  effective  actions  with  which  we  are  concerned,  this 
case  constitutes  no  exception  to  the  preceding  statement. 

609.  In  the  most  important  practical  cases,  as  we  shall  see  later, 
those  namely  m  which  the  substance  is  either  '  isotropic,'  which  is 
sensibly  the  case  with  common  metallic  wires,  or  has  an  axis  of 
elastic  symmetry  along  the  length  of  the  piece,  one  of  tlie  three 
normal  axes  of  torsion  and  flexure  coincides  with  the  length  oi  the 
wire,  and  the  two  others  are  perpendicular  to  it ;  the  first  being  an 
axis  of  pure  torsion,  and  the  two  others  axes  of  pure  flexure.  Thus 
opposing  couples  round  the  axis  of  the  wire  twist  it  simply  without 
bending  it ,  and  opposing  couples  in  either  of  the  two  principal  planes 
of  flexure,  bend  it  into  a  circle. 

610.  In  the  more  particular  case  in  which  two  principal  rigidities 
against  flexure  art  equal,  every  plane  through  the  length  of  the  wire 
is  a  principal  plane  of  flexure,  and  the  rigidity  against  flexure  is  equal 
m  all.  This  is  clearly  the  case  with  a  common  round  wire,  or  rod,  or 
with  one  of  square  section.  It  can  be  shown  to  be  the  casj  for  a 
rod  of  isotropic  material  and  of  any  form  of  normal  section  which 
is  *  kinetically  symmetrical '  (§  239)  round  all  axes  in  its  plane  through 
its  centre  of  inertia. 

611.  In  this  case,  if  one  end  of  the  rod  or  wire  be  held  fixed,  and 
a  couple  be  applied  in  any  plane  to  the  other  end,  a  uniform  spiral 
form  will  be  produced  round  an  axis  perpendicular  to  the  plane 
of  the  couple.  The  lines  of  the  substance  parallel  to  the  axis  of 
the  spiral  are  not,  however,  parallel  to  their  original  positions:  and 
lines  traced  along  the  surface  of  the  wire  parallel  to  its  length  when 
straight,  become  as  it  were  secondary  spirals,  circling  round  the 
main  spiral  formed  by  the  central  line  of  the  deformed  wire.  Lastly, 
in  the  present  case,  if  we  suppose  the  normal  section  of  the  wire 
to  be  circular,  and  trace  uniform  spirals  along  its  surface  when 
deformed  in  the  manner  suj,i]josed  (two  of  which,  for  instance,  are 
the  lines  along  which  it  is  touched  by  the  inscribed  and  the  circum- 


228  ABSTRACT  DYNAMICS. 

scribed  cylinder),  these  lines  do  not  become  straight,"  'but  become 
spirals  laid  on  as  it  were  round  the  wire,  when  it  is  allowed  to  take 
its  natural  straight  and  untwisted  condition. 

612.  A  wire  of  equal  flexibility  in  all  directions  may  clearly  be  Held 
in  any  specified-  spiral  form,  and  twisted  to  any  stated  degree,  by  a 
determinate  force  and  couple  applied  at  one  end,  the  other  end  being 
held  fixed.  The  direction  of  the  force  must  be  parallel  to  the  axis 
of  the  spiral,  and,  with  the  couple,  must  constitute  a  system  of  which 
this  line  is  (§  579)  the  central  axis  r  since  otherwise  there  could  not 
be  the  same  system  of  balancing  forces  in  every  normal  section  of 
the  spiral.  All  this  may  be  seen  clearly  by  supposing  the  wire  tO' 
be  first  brought  by  any  means  to  the  specified  condition  of  strain ; 
then  to  have  rigid  planes  rigidly  attached  to  its  two  ends  perpendicular 
to  its  axis,  and  these  planes  to  be  rigidly  connected  by  a  bar  lying 
in  this  line.  The  spiral  wire  now  left  to  itself  cannot  but  be  in 
equilibrium  :  although  if  it  be  too  long  (according  to  its  form  and 
degree  of  twist)  the  equilibrium  may  be  unstable.  The  force  along 
the  central  axis,  and  the  couple,  are  to  be  determined  by  the  condition 
that,  when  the  force  is  transferred  after  Poinsot's  manner  to  the  elastic- 
centre  of  any  normal  section,  they  give  two  couples  together  equiva- 
lent to  the  elastic  couples  of  flexure.and  torsion. 

613.  A  wire  of  equal  flexibility  in  all  directions  may  be  held  in 
any  stated  spiral  form  by  a  simple  force  along  its  axis  between  rigid 
pieces  rigidly  attached  to,  its  two  ends,  provided  that,  along  with  its 
spiral  form  a  certain  degree  of  twist  be  given  to  it.  The  force  is 
determined  by  the  condition  that  its  moment  round  the  perpendicular 
through  any  point  of  the  spiral  to  its  osculating  plane  at  that  point, 
must  be  equal  and  opposite  to  the  elastic  unbending  couple.  The 
degree  of  twist  is  that  due  (by  the  simple  equation  of  torsion)  to  the 
moment  of  the  force  thus  determined,  round  the  tangent  at  any  point 
of  the  spiral.  The  direction  of  the  force  being,  according  to  the 
preceding  condition,  such  as  to  press  together  the  ends  of  the  spiral) 
the  direction  of  the  twist  in  the  wire  is  opposite  to  that  of  the  tortuosity 
(§  13)  of  its  central  curve. 

614.  The  principles  with  which  we  have  just  been  occupied  are 
immediately  applicable  to  the  theory  of  spiral  springs ;  and  we 
shall  therefore  make  a  short  digression  on  this  curious  and  im- 
portant practical  subject  before  completing  our  investigation  of  elastic 
curves. 

A  common  spiral  spring  consists  of  a  uniform  ^re  shaped  per- 
manently to  have,  when  unstrained,  the  form  of  a  regular  helix,  with 
the  principal  axes  of  flexure  and  torsion  everywhere  similarly  situated 
relatively  to  the  curve.  When  used  in  the  proper  manner,  it  is 
acted  on,  through  arms  or  plates  rigidly  attached  to  its  ends,  by  forces 
such  that  its  form  as  altered  by  them  is  still  a  regular  helix.  This 
condition  is  obviously  fulfilled  if  (one  terminal  being  held  fixed)  an 


STATICS  OF  SOLIDS  AND  FLUIDS.  229 

infinitely  small  force  and  infinitely  small  couple  be  applied  to  the 
other  terminal  along  the  axis,  and  in  a  plane  perpendicular  to  it,  and 
if  the  force  and  couple  be  increased  to  any  degree,  and  always  kept 
along  and  in  the  plane  perpendicular  to  the  axis  of  the  altered  spiral. 
It  would,  however,  introduce  useless  complication  to  wofk  out  the 
details  of  the  problem  except  for  the  case  (§  609)  in  which  one  of 
the  principal  axes  coincides  with  the  tangent  to  the  central  line,  and 
is  therefore  an  axis  of  pure  torsion,  as  spiral  springs  in  practice 
always  belong  to  this  case.  On  the  other  hand,  a  very  interesting 
complication  occurs  if  we  suppose  (what  is  easily  realized  in  practice, 
though  to  be  avoided  if  merely  a  good  spring  is  desired)  the  normal 
section  of  the  wire  to  be  of  such  a  figure,  and  so  situated  relatively 
to  the  spiral,  that  the  planes  of  greatest  and  least  flexural  rigidity 
are  oblique  to  the  tangent  plane  of  the  cylinder.  Such  a  spring  when 
acted  on  in  the  regular  manner  at  its  ends  must  experience  a  certain 
degree  of  turning  through  its  whole  length  round  its  elastic  central 
curve  in  order  that  the  flexural  couple  developed  may  be,  as  we 
shall  immediately  see  it  must  be,  precisely  in  the  osculating  plane  of 
the  altered  spiral.  All  that  is  interesting  in  this  very  curious  effect  is 
illustrated  later  in  full  detail  (§  624  of  our  larger  work)  in  the  case  of  an 
open  circular  arc  altered  by  a  couple  in  its  own  plane,  into  a  circular 
arc  of  greater  or  less  radius ;  and  for  brevity  and  simplicity  we  shall 
confme  the  detailed  investigation  of  spiral  springs  on  which  we  now 
enter,  to  the  cases  in  which  either  the  wire  is  of  equal  flexural  rigidity 
in  all  directions,  or  the  two  principal  planes  of  (greatest  and  least  or 
least  and  greatest)  flexural  rigidity  coincide  respectively  with  the 
tangent  plane  to  the  cylinder,  and  the  normal  plane  touching  the 
central  curve  of  the  wire,  at  any  point. 

615.  The  axial  force,  on  the  movable  terminal  of  the  spring,  trans- 
ferred according  to  Poinsot  to  any  point  in  the  elastic  central 
curve,  gives  a  couple  in  the  plane  tiirough  that  point  and  the  axis 
of  the  spiral.  The  resultant  of  this  and  the  couj^le  which  we  suppose 
applied  to  the  terminal  in  the  plane  peqiendicular  to  the  axis  of  the 
spiral  is  the  effective  bending  and  twisting  couple  :  and  as  it  is  m  a 
plane  perj)endicular  to  the  tangent  plane  to  the  cylinder,  the  com- 
ponent of  it  to  which  bending  is  due  must  Ijc  also  perpendicular  to 
this  plane,  and  therefore  is  in  thf  osculating  plane  of  the  spiral.  Tins 
component  couple  therefore  simply  maintains  a  curvature  different 
from  the  natural  curvature  of  the  wire,  and  the  other,  tliat  is,  die 
couple  in  the  plane  normal  to  the  central  curve,  pure  tursiuu. 
The  equations  of  equilibrmm  merely  express  this  in  maihemat.ical 
language. 

616.  The  potential  energy  of  the  strained  spring  is 

I  [/.^(^-cJ'  +  ^^T-j/, 

if  A  denote  the  torsional  rigidity,  B  the  llexural  rigidity  in  tlie  plane 
of  curvature,  tcr  and  ct^  the  strained  and  unstrained  curvatures,  and  t 


230  ABSTRACT  DYNAMICS. 

the  torsion  of  the  wire  in  the  strained  condition,  the  torsion  being 
reckoned  as  zero  in  the  unstrained  condition.  The  axial  force,  and 
the  couple,  required  to  hold  the  spring  to  any  given  length  reckoned 
along  the  axis  of  the  spiral,  arid  to  any  given  angle  between  planes 
through  its  ends  and  the  axes,'  are  of  course  (§  244)' equal  to  the  rates 
of  variation  of  the  potential  energy,  per  unit  of  variation  of  these 
co-ordinates  respectively.  It  must  be  carefully  remarked,  however, 
that,  if  the  terminal  rigidly  attached  to  one  end  of  the  spring  be  held 
fast,  so  as  to  fix  the  tangent  at  this  end,  and  the  motion  of  the  other 
terminal  be  so  regulated  as  to  keep  the  figure  of  the  mtermediate 
spring  always  truly  spiral,  this  motion  will  be  somewhat  complicated; 
&s  the  radius  of  the  cylinder,  the  inclination  of  the  axis  of  the  spiral 
\o  the  fixed  direction  of  the  tangent  at  the  fixed  end,  and  the  position 
of  the  pomt  in  the  axis  in  which  it  is  cut  by  the  plane  perpendicular 
to  it  through  the  fixed  end  of  the  spnng,  all  vary  as  the  sprmg  changes 
in  figure.  The  effective  components  of  any  infinitely  small  motion  of 
the  movable  terminal  are  its  component  translation  along,  and  rota- 
tion round,  the  instantaneous  position  of  the  axis  of  the  spiral  [two 
degrees  of  freedom],  along  with  which  it  will  generally  have  -an 
infinitely  small  translation  m  some  direction  and  rotation  round  some 
line,  each  perpendicular  to  this  axis,  and  determined  from  the  two 
degress  of  arbitrary  motion,  by  the  condition  that  the  curve  remains  a 
true  spiral. 

617.  In  the  practical  use  of  spiral  springs,  this  condition  is  not 
rigorously  fulfilled :  but,  instead,  one  of  two  plans  is  generally  fol- 
lowed: — (i)  Force,  without  any  couple,  is  applied  pulling  out  or 
pressing  together  two  definite  points  of  the  two  terminals,  each  as 
nearly  as  may  be  in  the  axis  of  the  unstrained  spiral ;  or  {2)  One 
terminal  being  held  fixed,  the  other  is  allowed  to  slide,  without  any 
turning,  in  a  fixed  direction,  being  as  nearly  as  may  be  the  direction 
of  the  axis  of  the  spiral  when  unstrained.  The  preceding  investiga- 
tion is  applicable  to  the  infinitely  small  displacement  in  either  case  : 
the  couple  being  put  equal  to  zero  for  case  (1),  and  the  instantaneous 
rotatory  motion  round  the  axis  of  the  spiral  equal  to  zero  for 
case  (2). 

618.  In  a  spiral  spring  of  infinitely  small  inclination  to  the  plane 
perpendicular  to  its  axis,  the  displacement  produced  in  the  movable 
terminal  by  a  force  applied  to  it  in  the  axis  of  the  spiral  is  a  simple 
rectilineal  translation  in  the  direction  of  the  axis,  and  is  ecjual  to  the 
length  of  the  circular  arc  through  which  an  equal  force  carries  one 
end  of  a  rigid  arm  or  crank  equal  in  length  to  the  radius  of  the 
cylinder,  attached  perpendicularly  to  one  end  of  the  wire  of  the  spnng 
supposed  straightened  and  held  with  the  other  end  absolutely  fixed, 
and  the  end  which  bears  the  crank,  free  to  turn  in  a  collar.  This 
statement  is  due  to  J.  Thomson',  who  showed  that  in  pulling  out 
a  spiral  spring  of  infinitely  small  inclination  the  action  exercised  and 

1  Camb.  and  Dub.  Math.  Jour.  1848. 


STATICS   OF  SOLIDS  AND  FLUIDS.  231 

tlie  elastic  quality  used  are  the  same  as  in  a  torsion-balance  with  the' 
same  wire  straightened  (§  386).  This  theory  is,  as  he  proved  ex- 
perimentally, sufticiently  approximate  for  most  practical  applications; 
spiral  springs,  as  commonly  made  and  used,  being  of  very  small 
inclination.  There  is  no  difficulty  in  hnding  the  requisite  correcuon, 
for  the  actual  inclination  in  any  case.  The  fundamental  principle  that 
spiral  springs  act  chiefly  by  torsion  seems  to  have  been  first  discovered 
by  Binet  in  1814'. 

619.  R.eturning  to  the  case  of  a  uniform  wire  straight  and  untwisted 
(that  is,  cylindrical  or  prismatic)  when  free  from  stress;  let  us  suppose 
one  end  to  be  held  fixed  in  a  given  direction,  and  no  other  force 
from  without  to  influence  it  except  that  of  a  rigid  frame  attached  to 
its  other  end  acted  on  by  a  force,  R,  in  a  given  line,  AB,  and  a 
couple,  C,  in  a  plane  perpendicular  to  this  line.  The  form  and  twist 
it  will  have  when  in  equilibrium  are  determined  by  the  condition  that 
the  torsion  and  flexure  at  any  point,  /*,  of  its  length  are  those  due  to 
the  couple  G  compounded  with  the  couple  obtained  by  bringing  R 
Xo  P. 

620.  Kirchhoff  has  made  a  very  remarkable  comparison  between 
the  static  problem  of  bending  and  twisting  a  wire,  and  the  kinetic 
problem  of  the  rotation  ol  a  rigid  body.  We  can  give  here 
but  one  instance,  the  simplest  of  all — the  Elastic  Cune  of  James 
Bernoulli,  and  the  common  pendulum.  A  uniform  straight  wire, 
either  equally  flexible  in  all  planes  through  its  length,  or  having  its 
directions  of  maximum  and  mimmiun  flexural  rigidity  in  two  planes 
through  its  whole  length,  is  acted  on  by  a  force  and  couple  in  one 
of  these  planes,  applied  either  directly  to  one  end,  or  by  means  of  an 
arm  rigidly  attached  to  it,  the  other  end  being  held  fast.  The  force 
and  couple  may,  of  course  (§  568),  be  reduced  to  a  single  force,  the 
extreme  case  of  a  couple  being  mathematically  included  as  an  in- 
finitely small  force  at  an  infinitely  great  distance.  To  avoid  any 
restriction  of  the  problem,  we  must  suppose  this  force  applied  to  an 
arm  rigidly  attached  to  the  wire,  although  in  any  case  in  which  the 
line  of  the  force  cuts  the  wire,  the  force  may  be  applied  directly  at 
the  ••oint  of  intersection,  without  altering  the  circumstances  of  the 
wire'between  this  point  and  the  fixed  end.  The  wire  will,  in  these 
circumstances,  be  bent  into  a  curve  lying  throughout  in  the  plane 
through  Its  fixed  end  and  the  line  of  the  force,  and  (§  609)  its  curva- 
tures at  different  points  will,  as  was  first  shown  by  James  Bernoulli, 
be  simply  as  their  distances  from  this  line.  The  curve  fulfilling  this 
condition  has  clearly  just  two  independent  parameters,  of  which  one 
is  conveniently  regarded  as  the  mean  proportional,  a,  between  the 
radius  of  curvature  at  any  point  and  its  distance  Irom  the  line  of  force, 
and  the  other,  the  maximum  distance,  b,  of  the  wire  from  the  line  of 
force.  By  choosing  any  value  (or  each  of  these  parameters  it  is  easy 
to  trace  the  corresponding  curve  with  a  very  high  approximation  to 
accuracy,  by  commencing  with  a  small  circular  arc  touching  at  one 

*  St  Venant,  Comptcs  Rendus,  Sept.  1S64. 


syz  ABSTRACT  DYNAMICS. 

extremity  a  straight  line  at  the  given  maximum  distance  from  the  line 
of  force,  and  continuing  by  small  circular  arcs,  with  the  p*oper 
increasing  radii,  according  to  the  diminishing  distances  of  their  middle 
points  from  the  line  of  force.  The  annexed  diagrams  are,  however, 
not  so  drawn,  but  are  simply  traced  from  the  forms  actually  assumed 
by  a  flat  steel  spring,  of  small  enough  breadth  not  to  be  much 
disturbed  by  tortuosity  in  the  cases  in  which  different  parts  of  it  cross 
one  another.  The  mode  of  application  of  the  force  is  sufficiently 
explained  by  the  indications  in  the  diagram. 

621.  As  we  choose  particularly  the  common  pendulum  for  the 
corresponding  kinetic  problem,  the  force  acting  on  the  rigid  body  in 
the  comparison  must  be  that  of  gravity  in  the  vertical  through  its 
centre  of  gravity.  1^  is  convenient,  accordingly,  not  to  take  unity  as 
the  velocity  for  the  point  of  comparison  along  the  bent  wire,  but  the 
velocity  which  gravity  would  generate  in  a  body  falling  through  a  height 
equal  to  half  the  constant,  a,  of  §  620:  and  this  constant,  a,  will 
then  be  the  length  of  the  isochronous  simple  pendulum.  Thus  if 
an  elastic  curve  be  held  with  its  line  of  force  vertical,  and  if  a  point, 
|jP,  be  moved  along  it  with  a  constant  velocity  equal  to  Jga,  {a  de- 
noting the  mean  proportional  between  the  radius  of  curvature  at  any 
point,  and  its  distance  from  the  line  of  force,)  the  tangent  at  F  will 
keep  always  parallel  to  a  simple  pendulum,  of  length  a,  placed  at 
any  instant  parallel  to  it,  and  projected  with  the  same  angular 
velocity.  Diagrams  i  to  5,  correspond  to  vibratiotis  of  the  pendu- 
lum. Diagram  6  corresponds  to  the  case  in  which  the  pendulum 
would  just  reach  its  position  of  unstable  equilibrium  in  an  infinite 
time.  Diagram  7  corresponds  to  cases  in  which  the  pendulum  flies 
round  continuously  in  one  direction,  with  periodically  increasing  and 
diminishing  velocity.  The  extreme  case,  of  the  circular  elastic 
curve,'  corresponds  to  an  infinitely  long  pendulum  flying  round  with 
finite  angular  velocity,  which  of  course  experiences  only  infinitely 
small  variation  in  the  course  of  the  revolution.  A  conclusion  worthy 
of  remark  is,  that  the  rectification  of  the  elastic  curve  is  the  same 
analytical  problem  as  finding  the  time  occupied  by  a  pendulum  in 
describing  any  given  angle. 

622.  For  the  simple  and  important  case  of  a  natually  straight 
wire,  acted  on  by  a  distribution  of  force,  but  not  of  couple,  through 
jts  length,  the  condition  fulfilled  at  a  perfectly  frefi  end,  acted  on  by 
neither  force  nor  couple,  is  that  the  curvature  is  zero  at  the  end,  and. 
its  rate  of  variation  from  zero,  per  unit  of  length  from  the  end,  is, 
at  the  end,  zero.  Jn  other  words,  the  curvatures  at  points  infinitely 
near  the  end  are  as  the  squares  of  their  distances  from  the  end  in 
general  (or,  as  some  higher  power  of  these  distances,  in  singular 
cases).  The  same  statements  hold  for  the  change  of  curvature  pro- 
duced by  the  stress,  if  the  unstrained  wire  is  not  straight,  but  the 
Other  circumstances  the  same  as  those  just  specified. 

623.  As  a  very  simple  example  of  the  equiHbrium  of  a  wire  sub- 
ject to  forces  through  its  length,  let  us  suppose  the  natural  form  to 


STATICS  OF  SOLTDS  AND  FLUIDS 


Vol.  23—11 


234,  ABSTRACT  DYNAMICS. 

be  straight,  and  the  applied  forces  to  be  in  lines,  and  the  couples 
to  have  their  axes,  all  perpendicular  to  its  length,  and  to  be  not  great 
enough  to  produce  more  than  an  infinitely  small  deviation  from  the 
straight  line.  Further,  in  order  that  these  forces  and  couples  may 
produce  no  torsion,  let  the  three  flexure-torsion  axes  be  perpendicular 
to  and  along  the  wire.  But  we  shall  not  lirpit  the  problem  further 
by  supposing  the  section  of  the  wire  to  be  uniform,  as  we  should  thus 
exclude  some  of  the  most  important  practical  applications,  as  to 
beams  of  balances,  levers  in  machinery,  beams  iri  architecture  and 
engineering.  It  is  more  instructive  to  investigate  the  equations  of 
equilibrium  directly  for  this  case  than  to  deduce  them  from  the 
equations  worked  out  above  for  the  much  more  comprehensive 
general  .problem.  The  particular  principle  for  the  present  case  is 
sim.ply  that  the  rate  of  variation  of  the  rate  ef  variation,  per  unit  of 
length  along  the  wire,  of  the  bending  couple  in  any  plane  through 
the  length,  is  equal,  at  any  point,  to  the  applied  force  per  unit  of 
length,  with  the  simple  rate  of  variation  of  the  applied  couple  sub- 
tracted. This,  together  with  the  direct  equations  (§  607)  between 
the  component  bending  couples,  gives  the  required  equations  of 
equilibrium. 

624.  If  the  directions  of  maximum  and  minimum  flexural  rigidity 
lie  throughout  the  wire  in  two  planes,  the. equations  of  equilibrium 
become  simplified  when  these  planes  are  chosen  as  planes  of  re- 
ference, XOY,  XOZ.  The  flexure  in  either  plane  then  depends 
simply  on  the  forces  in  it,,  and  thus  the  problem  divides  itself  into 
two  quite  independent  problems  of  integrating  the  equations  of 
flexure  in  the  two  principal  planes,  and  so  finding  the  projections 
of  the  curve  on  two  fixed  planes  agreeing  with  their  position  when 
the  rod  is  straight. 

625.  When  a  uniform  bar,  beam,  or  plank  is  balanced  on  a  single 
trestle  at  its  middle,  the  droop  of  its  ends  is  only  -|  of  the  droop 
which  its  middle  has  when  the  bar  is  supported  on  trestles  at  its 
ends.  -From  this  it  follows  that  the  former  is  |  and  latter  f  of  the 
droop  or  elevation  produced  by  a  force  equal  to  half  the  weight  of 
the  bar,  applied  vertically  downwards  or  upwards  to  one  end  of  it, 
if  the  middle  is  held  fast  in  a  horizontal  position.  For  let  us  first 
suppose  the  whole  to  rest  on  a  trestle  under  its  middle,  and  let  two 
trestles  be  placed  under  its  ends  and  gradually  raised  till  the  pressure 
is  entirely  taken  off  from  the  middle.  During  this  operation  the 
middle  remains  fixed  and  horizontal,  while  a  force  increasing  to  half 
the  weight,  applied  vertically  upwards  on  each  end,  raises  it  through 
a  height  equal  to  the  sum  of  the  droops  in  the  two  cases  above 
referred  to.  This  result  is  of  course  proved  directly  by  comparing 
the  absolute  values  of  the  droop  in  those  two  cases  as  found  above, 
with  the  deflection  from  the  tangent  at  the  end  of  the  cord  in  the 
elastic  curve,  Fig.  2,  §  623,  which  is  cut  by  the  cord  at  right  angles. 
It  may  be  stated  otherwise  thus :  the  droop  of  the  middle  of  a 
uniform   beam   resting  on  trestles  at  its  ends  is  mcreased  in  the 


STATICS   OF  SOLIDS  AXD   FLUIDS.  235 

ratio  of  5  to  13  by  laying  a  mass  equal  in  weight  to  itself  on  its 
middle:  and,  if  the  beam  is  hung  by  its  middle,  the  droop  of  the 
ends  is  increased  in  the  ratio  of  3  to  11  by  hanging  on  each  of 
them  a  mass  equal  to  half  the  weight  of  the  beam. 

626.  The  important  practical  problem  of  finding  the  distribution 
of  the  weight  of  a  solid  on  points  supporting  it,  when  more  than 
two  of  these  are  in  one  vertical  plane,  or  when  there  are  more  than 
three  altogether,  which  (§  588)  is  indeterminate^  if  the  solid  is 
perfectly  rigid,  may  be  completely  solved  for  a  uniform  elastic  beam, 
naturally  straight',-  resting  on  three  or  more  points  in  rigorously  fixed 
positions  all  nearly  in  one  horizontal  line,  by  means  of  the  preceding 
results. 

If  there  are  /  points  of  support,  the  i-  1  parts  of  the  rod  between 
them  in  order  and  the  two  end  parts  will  form  z  +  i  curves  expressed 
by  distinct  algebraic  equations,  each  involving  four  arbitrary  con 
stants.  For  determining  these  constants  we  have  4^+4  equations  in 
all,  expressing  the  following  conditions: — 

I.  The  ordinates  of  the  inner  ends  of  the  projecting  parts  of  the 
rod,  and  of  the  two  ends  of  each  intermediate  part,  are  respectively 
equal  to  the  given  ordinates  of  the  corresponding  points  of  support 
[2/  equations]. 

II.  The  curves  on  the  two  sides  of  each  support  have  coincident 
tangents  and  equal  curvatures  at  the  point  of  transition  from  one 
to  the  other  [2/  equations]. 

III.  The  curvature  and  its  rate  of  variation  per  unit  of  length 
along  the  rod,  vanish  at  each  end  [4  equations]. 

Thus  the  equation  of  each  part  of  the  curve  is  completely  de- 
termined: and  by  means  of  it  we  find  the  shearing  force  in  any 
normal  section.  The  difference  between  these  in  the  neighbouring 
portions  of  the  rod  on  the  two  sides  of  a  point  of  support,  is  of 
course  equal  to  the  pressure  on  this  point. 

627.  The  solution  for  the  case  of  this  problem  in  which  two 
of  the  points  of  support  are  at  the  ends,  and  the  third  midway 
between  them  either  exactly  in  the  line  joining  them,  or  at  any 
given  very  small  distance  above  or  below  it,  is  found  at  once,  without 
analytical  work,  from  the  particular  results  stated  in  §  625.  Thus 
if  we  suppose  the  beam,  after  being  first  supported  wholly  by  trestles 
at  its  ends,  to  be  gradually  pressed  up  by  a  trestle  under  its 
middle,  it  will  bear  a  force  simply  proportional  to  the  space  through 
which  it  is  raised  from  the  zero  point,  until  all  the  weight  is  taken, 
off  the  ends,  and  borne  by  the  middle.     The  whole  distance  through 

which  tlie  middle  rises  during  this  process  is,  as  we  found, "^-^  .  —^ 

°        '■  J>    16.24. 

and  this  whole  elevation  is  f   of  the  droop  of  the  middle  in  the 

1  It  need  scarcely  be  remarked  that  indeterminatencss  does  not  exist  in  nature. 
How  it  may  occur  in  the  problems  of  abstract  dynamics,  and  is  obviated  by  tnkini; 
something  more  of  the  properties  of  matter  into  account,  is  instructively  illuitrated 
by  the  circumstances  referred  to  in  the  text. 


236  ABSTRACT  DYNAMICS. 

first  position.  If  therefore,  for  instance,  the  middle  trestle  be  fixed 
exactly  in  the  line  joining  those  under  the  ends,  it  will  bear  ■§■  of 
the  whole  weight,  and  leave  j\  to  be  borne  by  each  end.  And 
if  the  middle  trestle  be  lowered  from  the  line  joining  the  end 
ones  by  -/g-  of  the  space  through  which  it  would  have  to  be  lowered 
to  relieve  itself  of  all  pressure,  it  will  bear  just  -^  of  the  whole  weighty 
and  leave  the  other  two-thirds  to  be  equally  borne  by  the  two  ends., 

628.  A  wire  of  equal  flexibility  in  all  directions,  and  straight 
when  freed  from  stress,  offers,  when  bent  and  twisted  in  any  manner 
whatever,  not  the  slightest  resistance  to  being  turned  round  its  elastic 
central  curve,  as  its  conditions  of  equilibrium  are  in  no  way  affected 
by  turning  the  whole  wire  thus  equally  throughout  its  length..  The 
useful  application  of  this  principle,  to  the  maintenance  of  equal 
angular  motion  in  two  bodies  rotating  round  different  axes,  is 
rendered  somewhat  difficult  in  practice  by  the  necessity  of  a  perfect 
attachment  and  adjustment  of  each  end  of  the  wire,  so  as  to  have 
the  tangent  to  its  elastic, ;ceniTal  curve  exactly  in  line  with  the 
axis  of  rotation.  But  if  this  condition  is  rigorously  fulfilled,  and 
the  wire  is  of  exactly  equal  flexibility  in  every  direction,  and  exactly 
straight  when  free  from  stress,  it  will  give,  against  any  constant 
resistance,  an  accurately  uniform  motion  from  one  to  another  of 
two  bodies  rotating  round  axes  which  may  be  inclined  to  one 
another  at  any  angle,  and  need  not  be  in  one  plane.  If  they  are 
in  one  plane,  if  there  is  no  resistance  to  the  rotatory  motion,  and 
if  the  action  of  gravity  on  the  wire  is  insensible,  it  will  take  some 
of  the  varieties  of  form.  (§  620)  of  the  plane  elastic  curve  of  James 
Bernoulli.  But  however  much  it  is  altered  from  this,  whether  by 
the  axes  not  being  in  one  plane,  or  by  the  torsion  accompanying 
the  transmission  of  a  couple  from  one  shaft  to  the  other,  and 
necessarily,  when  the  axes  are  in  one  plane,  twisting  the  wire  out 
of  it,  or  by  gravity,  the  elastic  central  curve  will  remain  at  rest, 
the  wire  in  every  normal  section  rotating  round  it  with  uniform 
angular  velocity,  equal  to  that  of  each  of  the  two  bodies  which  it 
connects.  Under  Properties  of  Matter,  we  shall  see,  as  indeed 
may  be  judged  at  once  from  the  performances  of  the  vibrating 
spring  of  a  chronometer  for  twenty  years,  that  imperfection  in  the 
elasticity  of  a  metal  wire  does  not  exist  to  any  such  degree  as  to 
prevent  the  practical  application  of  this  principle,  even  in  mechanism 
required  to  be  durable. 

It  is  right  to  remark,  however,  that  if  the  rotation  be  too  rapid,  the 
equilibrium  of  the  wire  rotating  round  its  unchanged  elastic  central 
curve  may  become  unstable,  as  is  immediately  discovered  by  experi- 
ments (leading  to  very  curious  phenomena),  when,  as  is  often  done  in 
illustrating  the  kinetics  of  ordinary  rotation,  a  rigid  body  is  hung  by 
a  steel  wire,  the  upper  end  of  which  is  kept  turning  rapidly. 

629.  The  definitions  and  investigations  regarding  strain,  §§  135- 
161,  constitute  a  kincmatical  introduction  to  the  theory  of  elastic 
solids.     We  must  now,  in  commencing   the   elementary  dynamics 


STATICS  OF  SOLIDS  AND  FLUIDS, 


237 


of  the  subject,  consider  the  forces  called  into  play  through  the 
interior  of  a  solid  when  brought  into  a  condition  of  strain.  We 
adopt,  from  Rankine',  the  term  stress  to  designate  such  forces, 
as  distinguished  from  strain  defined  (§  135)  to  express  the  merely 
geometrical  idea  of  a  change  of  volume  or  figure. 

630.  When  through  any  space  in  a  body  under  the  action  of  force, 
the  mutual  force  between  the  portions  of  matter  on  the  two  sides  of 
any  plane  area  is  equal  and  parallel  to  the  mutual  force  across  any 
equal,  similar,  and  parallel  plane  area,  the  stress  is  said  to  be  homo- 
geneous through  that  space.  In  other  words,  the  stress  experienced 
by  the  matter  is  homogeneous  through  any  space  if  all  equal,  similar, 
and  similarly  turned  portions  of  matter  within  this  space  are  similarly 
and  equally  influenced  by  force. 

631.  To  be  able  to  find  the  distribution  of  force  over  the  surface- 
of  any  portion  of  matter  homogeneously  stressed,  we  must  know  the 
direction,  and  the  amount  per  unit  area,  of  the  force  across  a  plane 
area  cutting  through  it  in  any  direction.  Now  if  we  know  this  for 
any  three  planes,  in  three  different  directions,  we  can  find  it  for  a 
plane  in  any  direction  as  we  see  in  a  moment  by  considering  what 
IS  necessary  for  the  equilibrium  of  a  tetrahedron  of  the  substance.  The 
resultant  force  on  one  of  its  sides  must  be  equa'l  and  opposite  to  the 
resultant  of  the  forces  on  the  three  otherSj  which  is  known  if  these  sides 
are  parallel  to  the  three  planes  for  each  of  which  the  force  is  given. 

632.  Hence  the  stress,  in  a  body  homogeneously  stressed,  is  com- 
pletely specified  when  the  direction,  and  the  amount  per  unit  area, 
of  the  force  on  each  of  three  distinct  planes  is  given.  It  is,  in  the 
analytical  treatment  of  the  subject,  generally  convenient  to  take  these' 
planes  of  reference  at  right  angles  to  one  another.  But  we  should 
immediately  fall  into  error  did  we  not  remark  that  the  specification- 
here  indicated  consists  not  of  nine  but  in  reality  only  of  six,  inde- 
pendent elements.  For  if  the  equilibrating  forces  on  the  six  faces  of 
a  cube  be  each  resolved  into  three  components  parallel  to  its  three 
edges,    OX,  OV,   OZ,   we  Jiave  in 

all  18  forces;  of  which  \each  pair 
acting  perpendicularly  on 'a'  pair  of 
opposite  faces,  being  equal  and 
directly  opposed,  balance  one  an- 
other. The  twelve  .tangential  com- 
ponents that  remain  constitute  three 
pairs  of  couples  having  their  axes 
in  the  direction  of  the  three  edges, 
each  of  wliich  must  separately  be  in  T 
equilibrium.      The    diagram    shows  Q'- 

the  pair  of  equilibrating  couples 
having  OV  for  axis;  from  the  con- 
sideration of  which  we  infer  that  the  ^ 

^  Cambruigs  oHi:!  LH(5lin  ^laihinialical  j\  lur.a.',  iSjO. 


z\ 


-!:> 


X 


238  ABSTRACT  DYNAMICS. 

forces  on  the  faces  {zy),  parallel  to  OZ^  are  equal  to  the  forces  on  the 
faces  {yx)^  parallel  to  OX.  Similarly,  we  see  that  the  forces  on  the 
faces  {yx),  parallel  to  OY,  are  equal  to  those  on  the  faces  {xz),  parallel 
to  OZ;  and  that  the  forces  on  {xz),  parallel  to  OX,  are  equal  to 
those  on  {zy),  parallel  to  O  Y. 

633.  Thus,  any  three  rectangular  planes  of  reference  being  chosen, 
we  may  take  six  elements  thus,  to  specify  a  stress :  P,  Q,  R  the 
normal  components  of  the  forces  on  these  planes;  and  S.,  T,  U 
the  tangential  components,  respectively  perpendicular  to  OX,  of  the 
forces  on  the  two  planes  meeting  in  OX,  perpendicular  to  OY,  of 
the  forces  on  the  planes  meeting  \v\.  OY,  and  perpendicular  to  OZ, 
of  the  forces  on  the  planes,  meeting  in  OZ;  each  of  the  six  forced 
being  reckoned,  per  unit  of  area.  A  normal  component  will  be 
reckoned  as  positive  when  it  is  a  traction  tending  to  separate 
the  portions  of  matter  on  the  two  sides  of  its  plane.  P,  Q,  R  are 
sometimes  called  simple  longitudinal  stresses,  and  S,  T,  U  simple 
shearing  stresses. 

From  these  data,  to  find  in  the  manner  explained  in  §  631,  the 
force  on  any  plane,  specified  by  /,  ni,  n,  the  direction-cosines  of 
its  normal ;  let  such  a  plane  cut  OX,  OY,  OZ  in  the  three  points 
X,  Y,  Z.  Then,  if  the  area  XYZ  be  denoted  for  a  moment  by  A, 
the  areas  YOZ,  ZOX,  XO  Y,  being  its  projections  on  the  three  rec- 
tangular planes,  will  be  respectively  equal  to  A/,  Am,  An.  Hence, 
for  the  equilibrium  of  the  tetrahedron  of  matter  bounded  by  those 
four  triangles,  we  have,  ifi^,  G,  J7  denote  the  components  of  the 
force  experienced  by  the  first  of  them,  XYZ,  per  unit  of  its  area, 

F.A=P.IA+  U.mA+T.nA, 
and  the  two  symmetrical  equations  for  the  components  parallel  to 
OYzxid.  OZ.     Hence,  dividing  by  A,  we  conclude 

F  =  Pl  +  Um^Tn) 

G^Ul+Qm  +  Sn\,  (i) 

H=  TI+  Sm  +  Rn) 
These  expressions  stand  in  the  well-known  relation  to  the  ellipsoid 

Px^  ^Qf^-Rz^+z  {Syz  +  Tzx  +  Uxy)  =  1,  (2) 

according  to  which,  if  we  take 

X  =  /r,  y-  mr,  z^nr^ 
and  if  X,  \i,  v  denote  the  direction-cosines  and  p  the  length  of  the 
perpendicular  from  the  centre  to  the  tangent  plane  at  {x,  y,  z)  of  the 
ellipsoidal  surface,  we  have 

/r=A,     0=^^,     H=^. 

pr  pr  pr 

We  conclude  that 

631  For  any  fully  specified  state  of  stress  in  a  solid,  a  quadratic 
surface  may  always  be  determined,  which  shall  represent  the  stress 
graphically  in  the  following  manner: — 

To  find  the  direction  and  the  amount  per  unit  area,  of  the  forc$ 


STATICS   OF  SOLIDS  AND   FLUIDS.  259 

acting  across  any  plane  in  the  solid,  draw  a  line  perpendicular  to 
this  plane  from  the  centre  of  the  quadratic  to  its  surface.  The 
required  force  will  be  equal  to  the  reciprocal  of  tlie  product  of  the 
length  of  this  line  into  the  perpendicular  from  the  centre  to  the 
tanL,cnt  plane  at  the  point  of  intersection,  and  will  be  perpendicular 
to  the  latter  plane. 

635.  From  this  it  follows  that  for  any  stress  whatever  there  are 
three  determinate  planes  at  right  angles  to  one  another  such  that  the 
force  acting  in  the  solid  across  each  of  them  is  precisely  perpendicular 
to  it.  These  planes  are  called  the  principal  or  normal  planes  of  the 
stress;  the  forces  upon  them,  per  unit  area, — its  principal  or  normal 
tractions;  and  the  lines  perpendicular  to  them, — its  principal  or 
normal  axes,  or  simply  its  axes.  The  three  principal  semi-diameters 
of  die  quadratic  surface  are  equal  to  the  reciprocals  of  the  square 
roots  of  the  normal  tractions.  If,  however,  in  any  case  each  of  the 
three  normal  tractions  is  negative,  it  will  be  convenient  to  reckon 
them  rather  as  positive  pressures ;  the  reciprocals  of  the  square  roots 
of  which  will  be  the  semi-axes  of  a  real  stress-ellipsoid  representing 
the  distribution  of  force  in  the  manner  explained  above,  with  pressure 
substituted  throughout  for  traction. 

636.  When  the  three  normal  tractions  are  all  of  one  sign,  the 
stress-quadratic  is  an  ellipsoidj  the  cases  of  an  ellipsoid  of  revolution 
and  a  sphere  being  included,  as  those  in  which  two,  or  all  three,  are 
equal.  When  one  of  the  three  is  negative  and  the  two  others  posi- 
tive, the  surface  is  a  hyperboloid  of  one  sheet  When  one  of  the 
normal  tractions  is  positive  and  the  two  others  negative,  the  surface 
is  a  hyperboloid  of  two  sheets. 

637.  When  one  of  the  three  principal  tractions  vanishes,  while 
the  other  two  are  finite,  the  stress-quadratic  becomes  a  cylinder, 
circular,  elliptic,  or  hyperbolic,  according  as  the  other  two  are  equal, 
unequal  of  one  sign,  or  of  contrary  signs.  When  two  of  the  three 
vanish,  the  quadratic  becomes  two  planes;  and  the  stress  in  this  case 
is  (§  633)  called  a  simple  longitudinal  stress.  The  theory  of  prin- 
cipal planes,  and  normal  tractions  just  stated  (§  635),  is  then  equiva- 
lent to  saying  that  any  stress  whatever  may  be  regarded  as  made  up 
of  three  simple  longitudinal  stresses  in  three  rectangular  directions. 
The  geometrical  interpretations  are  obvious  in  all  these  cases. 

638.  The  composition  of  stresses,  is  of  course  to  be  effected  by 
adding  the  component  tractions  thus: — If  (/',,  Q^,  F^,  S^,  Z",,  d^), 
{F..,  Q„,  F^,  S,,  T^,  U^,  etc.,  denote,  according  to  §  633,  any  given 
set  of  stresses  acting  simultaneously  in'  a  substance,  their  joint  effect 
is  the  same  as  that  of  a  simple  resultant  stress  of  v.-hich  the  specifica- 
tion in  corresponding  terms  is  (S/',  ^Q,  2A',  "S^S,  27",  26^. 

639.  Each  of  the  statements  that  have  now  been  made  (j§  630, 
63 S)  regarding  stresses,  is  applicable  to  injhiitely  small  strains,  if  for 
traction  perpendicular  to  any  plane,  reckoned  per  unit  of  its  area, 
we  substitute  elongation,  in  the  lines  of  the  traction,  reckoned  per 
unit  of  length;  and  for  half  the  tangential  traction   parallel   to   any 


240 


ABSTRACT  DYNAMICS. 


direction,  shear  in  the  same  direction, -reckoned  in  the  manner  ex- 
plained in  §  154.  The  student  will  find  it  a  useful  exercise  to  study 
in  detail  this  transference  of  each  one  of  those  statements,  and  to 
justify  it  by  modifying  in  the  proper  manner  the-results  of  §§  150,  151, 
^52,  153,  154,  161,  to  adapt  them  to  infinitely  small  strains.  It 
must  be  remarked  that  the  strain-quadratic  thus  formed  according  to 
the  rule  of  §  634,  which  niay  have  any  of  the  varieties  of  character 
mentioned  in  §§  62,6,  637,  is  not  the  same  as  the  strain-ellipsoid  of 
§  141,  which  is  always  essentially  an  ellipsoid,  and  which,  for  an  in- 
finitely small  strain,  differs  infinitely  little  from  a  sphere. 

The  Comparison  of  §"151,  with  the  result  of  §  632  regarding  tan- 
gential tractions  is  particularly  interesting  and  important 

640.  The  following  tabular  synopsis  of  the  meaning  of  the 
elements  constituting  the  corresponding  rectangular  specifications  of 
.a  strain  and  qtress  explained  in  preceding  sections,  will  be  found 
convenient: — 

Components 

of  the 

strain,      stress. 


t  P 

f        Q 
s 

T 
U 


641.     If  a  unit  cube  of  matter  under  any  stress  {P,  Q,  R,  S,  T,  U) 

experience -the  infinitely  small  simple  longitudinal  strain  «?  alone,  the 
work  done  on  it  will  be  Fe;  since,  of  the  component  forces,  P,  U,  T 
parallel  to  (9 A",  f/and  T'do  no  work  in  virtue  of  this  strain.  Simi- 
larly, (2/,  Rg  are  the  works  done  if,  the' same  stress  acting,  the  simple 
longitudinal  strains  f  or  g  are  experienced,  either  alone.  Again,  if 
the  cube  experiences  a  simple  shear,  a,  whether  we  regard  it  (§  151) 
as  a  differential  sliding  of  the  planes  ^.r,  parallel  to^,  or  of  the  planes 
zx,  parallel  to  ^,  we  see  that  the  work  done  is  Sa:  and  similarly, 
Tb  if  the  strain  is  simply  a  shear  b,  parallel  to  OZ,  of  planes  zy,  or 
parallel  to  OX,  of  planes  xy:  and  Uc  if  the  strain  is  a  shear  c,  parallel 
to  OX,  of  planes  xz^  or  parallel  to  OY,  of  planes  yz.  Hence  the 
whole  work  done  by  the  stress  {P,  Q,  R,  S,  T,  (7)  on  a  unit  cube 
taking  the  strain  {e,  /,  g,  a,  b,  c),  is 

Pg+ Q/+ Rg  + Sa+  77>  +  [/c.  (3) 

It  is  to  be  remarked  that,  inasmuch  as  the  action  callec}  a  stress  is 
a  system  of  forces  which  balance  one  another  if  the  'portion  of 
matter  experiencing  it  is  rigid,  it  cannot  do  any  work  when  the 


Planes ;  of  which 

Direction 

relative  motion,  or 

of  relative 

across  which   force 

motion  or 

is  reckoned. 

of  force. 

yz 

X 

zx 

y 

xy 

z 

Jyx 

y 

\zx 

z 

zy 

z 

[xy 

X 

<xz 

X 

[yz 

y 

STATICS   OF  SOLIDS   AND   FLUIDS.  241 

matter  moves  in  any  way  without  change  of  shape:  and  therefore  no 
amount  of  translation  or  rotation  of  the  cube  taking  place  along  with 
the  strain  can  render  the  amount  of  work  done  dilTcrcnt  from  that 
just  found. 

If  the  side  of  the  cube  be  of  any  length)^,  instead  of  unity,  each 
force  will  be /^  times,  and  each  relative  displacement/  times,  and, 
therefore,  the  work  done  p^  times  the  respective  amounts  reckoned 
above.  Hence  a  body  of  any  shape,  and  of  cubic  content  C,  sub- 
jected throughout  to  a  uniform  stress  (/",  Q,  R,  S,  T,  U)  while  taking 
uniformly  throughout  a  stram  {e,/,g,  a,  b,  c),  experienced  an  amount 
of  work  equal  to 

{Pe  +  (9/+  Fg^Sa^-  Tb  +  Uc)  C.  (4) 

It  is  to  be  remarked  that  this  is  necessarily  equal  to  the  work  done 
on  the  bounding  surface  of  the  body  by  forces  applied  to  it  from 
without.  For  the  work  done  on  any  portion  of  matter  within  the 
body  is  simply  that  done  on  its  surface  by  the  matter  touching  it  all 
round,  as  no  force  acts  at  a  distance  from  without  on  the  interior 
substance.  Hence  if  we  imagine  the  whole  body  divided  into  any 
number  of  parts,  each  of  any  shape,  the  sum  of  the  work  done  on 
ajl  these  parts  is,  by  the  disappearance  of  equal  positive  and  negative 
terms  ex[)res5ing  the  portions  of  the  work  done  on  each  part  by  the 
contiguous  parts  on  all  its  sides,  and  spent  by  these  other  parts  in 
this  action,  reduced  to  the  integral  amount  of  work  done  by  force 
from  without  applied  all  round  the  outer  surface. 

642.  If,  now,  we  suppose  the  body  to  yield  to  a  stress  {P,  Q,  R, 
S,  T,  U),  and  to  oppose  this  stress  only  with  its  innate  resistance  to 
change  of  shape,  the  differential  equation  of  work  done  will  [by  (4) 
with  de,  d/,  etc.,  substituted  for  e, /,  etc.]  be 

dco  =  Pde  +  Qdf  +  Rdg  +  Sda  +  Tdb  +  Udc.  (5 ) 

If  w  denote  the  whole  amount  of  \vork  done  per  unit  of  volume  in 
any  part  of  the  body  while  the  substance  in  this  part  experiences  a 
strain  (e,f,  g,  a,  b,  r)  from  some  initial  state  regarded  as  a  state  of 
no  strain.  This  equation,  as  we  shall  see  later,  under  Properties  of 
Matter,  expresses  the  work  done  in>a  natural  fluid,  by  distorting 
stress  (or  difference  of  pressure  in  diffei*fent  directions)  working  against 
its  innate  viscosity;  and  w  is  then,  according  to  Joule's  discover}-, 
the  dynamic  value  of  the  heat  generated  in  the  process.  The  equa- 
tion may  also  be  applied  to  express  the  work  done  in  straining  an 
imperfectly  elastic  solid,  or  an  elastic  solid  of  which  the  temperature 
varies  during  the  process.  In  all  such  applications  the  stress  will 
depend  partly  on  the  speed  of  the  straining  motion,  or  on  the  var}-ing 
temperature,  and  not  at  all,  or  not  solely,  on  the  state  of  strain  at  any 
moment,  and  the  system  will  not  be  dynamically  conservative. 

643.  Dcfinition.r—K  perfectly  elastic  body  is  a  body  which,  when 
brought  to'  any  one  state  of  strain,  requires  at  all  times  the  same 
stress  to  hold  it  in  this  state;  liowever  long  it  be  kept  strained,  or 
however  rapidly  its  state  be  altered  from  any  other  strain,  or  from 
no  strain,  to  the  strain  in  question.     Here,  according  to  our  plan 


242  ABSTRA  CT  D  YNAMICS. 

(§§  39^)  4°')  ^or  Abstract  Dynamics,  we  ignore  variation  of  tempera- 
ture in  the  body.  If,  however,  we  add  a  condition  of  absolutely  no 
variation  of  temperature,  or  of  recurrence  to  one  specified  temperature 
after  changes  of  strain,  we  have  a  definition  of  that  property  of  perfect 
elasticity  towards  which  highly  elastic  bodies  in  nature  approximate; 
and  which  is  rigorously  fulfilled  by  all  fluids,  and  may  be  so  by  some 
real  solids,  as  homogeneous  crystals.  But  inasmuch  as  the  elastic 
reaction  of  every  kind  of  body  against  straiii  varies  with  varying 
temperature,  and  (a  thermodynamic  consequence  of  this,  as  we  shall 
see  later)  any  increase  or  diminution  of  strain  in  an  elastic  body  is 
necessarily'  accompanied  by  a  change  of  temperature;  even  a  per- 
fectly elastic  body  could  not,  in  passing  through  different  strains, 
act  as  a  rigorously  conservative  system,  but  on  the  contrary,  must 
give  rise  to  dissipation  of  energy  in  consequence  of  the  conduction 
or  radiation  of  heat  induced  by  these  changes  of  temperature. 

But  by  making  the  changes  of  strain  quickly  enough  to  prevent 
any  sensible  equilization  of  temperature  by  conduction  or  radiation 
(as,  for  instance,  Stokes  has  shown,  is  done  in  sound  of  musical 
notes  travelling  through  air);  or  by  making  them  slowly  enough  to 
allow  the  temperature  to  be  maintained  sensibly  constant*  by  proper 
appliances;  any  highly  elastic,  or  perfectly  elastic  body  in  nature  may 
be  got  to  act  very  nearly  as  a  conservative  system. 

644.  In  nature,  therefore,  the  integral  amount,  w,  of  w'ork  defined 
as  above,  is  for  a  perfectly  elastic  body,  independent  (§  246)  of  the 
series  of  configurat^ions,  or  states  of  strain,  through  which  it  may  have 
been  brought  from  the  first  to  the  second  of  the  specified  conditions, 
provided  it  has  not  been  allowed  to  change  sensibly  in  temperature 
during  the  process. 

When  the  whole  amount  of  strain  is  infinitely  small,  and  the  stress- 
components  are  therefore  all  altered  in  the  same  ratio  as  the  strain- 
components  if  these  are  altered  all  in  any  one  ratio;  w  must  be  a 
homogeliieous  quadratic  function  of  the  six  variables  e,  /,  g,  a,  b,  c, 
which,  if  we  denote  by  (e,  e),  {/,/). ..  (<?,/)•  •  .constants  depending 
on  the  quality  of  the  substance  and  on  the  directions  chosen  for  the 
axes  of  co-ordinates,  we  may  write  as  follows: — 

^=h{  (^.  ^)  ^  +  (//)/'  +  (i,^)  S'  +  (^, «)  «'  +  (^  ^)  b'  +  {c,  c)  ^ 

+  2  {e,/)ef+  2  {e,g)  e g  +  2  {e,  a)  e  a  +  2  {e,  b)  e  b  ■¥  2  (e,c)  ec 

+  2  (/ i^)/^  4- 2  (/ a /«  +  2  (/ ^)/^  +  2  (/ 0/^ 

+  2  {g,  a)ga  +  2  (^,  b)gb  +  2  (^,  c)gc 

+  2  (a,  b)ab  +  2  {a,c)  ac 

+  2  {b,c)bc\ 

The  21  co-eflicients  {e.,e),  {/,/)■.  »{b,c),  in  this  expression  con- 
stitute the  21  'co-efficients  of  elasticity,'  which  Green  first  showed  to 
be  proper  and  essential  for  a  complete  theory  of  the  dynamics  of  an 
elastic  solid  subjected  to  infinitely  small  strains.     The  only  condition 

*  'On  the  Thermoclastic  and  Thcrmomagnctic  Properties  of  Matter'  (W. 
Thomson).      Quarterly  Journal  of  ^lathematus,  h^\\\  x'^^l.  ^  Ibid. 


STATICS   OF  SOLIDS  AND  FLUIDS.  243 

that  can  be  theoretically  imposed  upon  these  co-efficients  is  that  they 
must  not  permit  w  to  become  negative  for  any  values,  positive  or 
negative,  oi  the  strain-components  e, /, . . .  c.  Under  Properties  of 
Matter,  we  shall  see  that  a  false  theory  (Boscovieh's),  falsely  worked 
out  by  mathematicians,  has  led  to  relations  among  the  co-efficients  "of 
elasticity  which  experiment  has  proved  to  be  false. 

645.  The  average  stress,  due  to  elasticity  of  the  solid,  when 
strained  from  its  natural  condition  to  that  of  strain  {e,f,  g,  a,  b,  c)  is 
(as  from  the  assumed  applicability  of  the  principle  of  superposition 
we  see  it  must  be)  just  half  the  stress  required  to  keep  it  in  this 
state  of  strain. 

646.  A  body  is  called  homogeneoas  when  any  two  equal,  similar 
parts  of  it,  with  corresponding  lines  parallel  and  turned  towardi 
the  same  parts,  are  undistinguishable  from  one  another  by  any 
difference  in  quality.  The  perfect  fulfilment  of  this  condition  with- 
out any  limit  as  to  the  sraallness  of  the  parts,  tJiough  conceivable, 
is  not  generally  regarded  as  probable  for  any  of  the  real  solids  or 
fluids  known  to  us,  however  seemingly  homogeneous.  It  is,  we 
believe,  held  by  all  naturalists  that  there  is  a  molecular  structure, 
according  to  which,  in  compound  bodies  such  as  water,  ice,  rock- 
crystal,  etc.,  the  constituent  substances  lie  side  by  side,  or  arranged 
in  groups  of  finite  dimensions,  and  even  in  bodies  called  simple 
(i.e.  not  known  to  be  chemically  resolvable  into  other  substances) 
there  is  no  ultimate  homogeneousness.  In  other  words,  the  prevail- 
ing belief  is  that  every  kind  of  matter  with  which  we  are  acquainted 
has  a  more  or  less  coarse-grained  texture,  whether  having  visible 
molecules,  as  great  masses  of  solid  stone  or  brick-building,  or  natural 
granite  or  sandstone  rocks;  or,  molecules  too  small  to  be  visible 
or  directly  measurable  by  us  (but  not  infinitely  small) '  in  seemingly 
homogeneous  metals,  or  continuous  crystals,  or  hquids,  or  pases. 
We  must  of  course  return  to  this  subject  under  Properties  oi  Matter ; 
and  in  the  meantime  need  only  say  that  the  definition  of  homogeneous- 
ness may  be  applied  practically  on  a  very  large  scale  to  masses  of 
building  or  coarse-grained  conglomerate  rock,  or  on  a  more  moderate 
scale  to  blocks  of  common  sandstone,  or  on  a  very  small  scale  to 
seemingly  homogeneous  metals';  or  on  a  scale  of  extreme,  undis- 
covered fineness,  to  vitreous  bodies,  continuous  crystals,  solidilied 
gums,  as  India  rubber,  gum-arabic,  etc.,  and  fluids. 

647.  The  substance  of  a  homogeneous  solid  is  called  isotropic 
when  a  spherical  portion  of  it,  tested  by  any  physical  agency,  exhibits 
no  difference  in  quality  however  it  is  turned.  Or,  which  amounts 
to  the  same,  a  cubical  portion  cut  from  any  position  in  an  isotropic 
body  exhibits  the  same  qualities  relatively  to  each  pair  of  parallel 
faces.     Or  two  equal   and   similar  portions  cut  from  any  positions 

^  Probably  not  undhcoznatly  im:i\\,  althoii;^h  of  dimensions  not  yet  kn>:wn  to  us. 
*  Which,  however,  we  know,  as  recently  proved  by  Deville  and  Van  Troost,  aie 
porous  enough  at  high  temperature  to  allow  very  free  percolation  of  gases. 


244  ABSTRACT  DYNAMICS. 

in  the  body,  not  subject  to  the  condition  of  parallelism  (§  646), 
are  undistinguishable  from  one  another.  A  substance  which  is  not 
isotropic,  but  exhibits  differences  of  quality  in  different  directions, 
is  called  aeolotropic. 

648.  An  individual  body,  or  the  substance  of  a  homogeneous 
solid,  may  be  isotropic  in  one  quality  or  class  of  qualities,  but 
aeolotropic  in  others. 

Thus  in  abstract  dynamics  a  rigid  body,  or  a  group  of  bodies 
rigidly  connected,  contained  within  and  rigidly  attached  to  a  rigid 
spherical  surface,  is  kinetically  symmetrical  (§  239)  if  its  centre  of 
inertia  is  at  the  centre  of  the  sphere,  and  if  its  moments  of  inertia 
are  equal  round  all  diameters. '  It  is  also  isotropic  relatively  to  gravi- 
tation if  it  is  centrobaric  (§  542),  so  that  the  centre  of  figure  is  not 
merely  a  centre  of  inertia,  but  a  true  centre  of  gravity.  Or  a  trans- 
parent substance  may  transmit  hght  at  different  velocities  in  different 
directions  through  it  (that  is,  be  dotibly  refracting),  and  yet  a  cube 
of  it  may  (and  generally  does  in  natural  crystals)  absorb,  the  same 
part  of  a  beam  of  white  light  transmitted  across  it  perpendicularly 
to  any  of  its  three  pairs  of  faces.  Or  (as  a  crystal  which  exhibits 
dichroism)  it  may  be  aeolotropic  relatively  to  the  latter,  or  to  either, 
optic  quality,  and  yet  it  may  conduct  heat  equally  in  all  directions. 

649.  The  remarks  of  §  64^  relative  to  homogeneousness  in  the 
aggregate,  and  the  supposed  ultimately  heterogeneous  texture  of  all 
substances  however  seemingly  homogeneous,  indicate  corresponding 
limitations  and  non-rigorous  practical  interpretations  of  isotropy. 

650.  To  be  elastically  isotropic,  we  see  first  that  a  spherical  or 
cubical  portion  of  any  solid,  if  subjected  to  uniform  normal  pressure 
(positive  or  negative)  all  round,  must,  in  yielding,  experience  no 
deformation:  and  therefore  must  be  equally  compressed  (or  dilated) 
in  all  directions.  But,  further,  a  cube  cut  from  any  position  in  it, 
and  acted  on  by  tangmtial  or  distorting  stress  (§  633)  in  planes 
parallel  to  two  pairs  of  its  sides,  must  experience  simple  deformation, 
or  shear  (§  150),  in  the  same  direction,  unaccompanied  by  condensa- 
tion or  dilatation',  and  the  same  in  amount  for  all  the  three  ways 
in  which  a  stress  may  be  thus  applied  to  any  one  cube,  and  for 
different  cubes  taken  from  any  different  positions  '\i\  the  solid. 

651.  Hence  the  elastic  quality  of  a  perfectly  elastic,  homogeneous,- 
isotropic  solid  is  fully  defined  by  two  elements; — its  resistance  to 
compression,  and  its  resistance  to  distortion,  f;^  The  amount  of  uni- 
form pressure  in  all  directions,  per  unit^area  of  its  surface,  required 
•to  produce   a  stated  very  small  compression,  measures  the  first  of 

^'  It  must  be  remembered  that  the  changes  of  figure  and  volume  we  are  con- 
^eerned  w^itl^arc  so  small  that  the  principle  of  superj^osition  is  applicable;  so  that 
if  any  distoTting  stress  produced  a  condensation,  an  opposite  distorting  stress 
would  produce  a  dilatation,  whichisa  violation  of  the  isotropic  condition.  But  it 
is  possible  that  a  distorting  stress  may  produce,  in  a  truly  isotropic  solid,  conden- 
sation or  dilatation  in  proportion  to  the  square  of  its  value:  and  it  is  probable  Iha^ 
'such  effects  may  be  sensible  in  India  rubber,  or  cork,  or  other  bodies  susceptible 
|of  great  deformations  or  compressions,  with  persistent  elasticity. 


STATICS   OF  SOLIDS  AND   FLUIDS. 


245 


these,  and  the  amount  of  the  distorting  stress  required  to  produce 
a  stated  amount  of  distortion  measures  the  second.  The  numerical 
measure  of  the  first  is  the  compressing  pressure  divided  by  the 
diminution  of  the  bulk  of  a  portion  of  the  substance  which,  when 
uncompressed,  occupies  the  unit  volume.  It  is  sometimes  called 
the  elasticity  of  volume,  or  the  resistance  to  compression.  Its  reciprocal, 
or  the  amount  of  compression  on  unit  of  volume  divided  by  tlie 
compressing  pressure,  or,  as  we  may  conveniently  say,  the  compression- 
per  unit  of  volume,  per  unit  of  compressing  pressure,  is  commonly 
called  the  compressibility.  The  second,  or  resistance  to  change  of  shape, 
is  measured  by  the  tangential  stress  (reckoned  as  in  §  633)  divided  by 
the  amount  of  the  distortion  or  shear  (§  154)  which  it  produces,  anc 
is  called  the  rigidity  of  the  substance,  or  its  elasticity  of  figure. 

652.  From  §  148  it  follows  that  a  strain  compounded  of  a  simple 
extension  in  one  set  of  parallels,  and  a  simple  contraction  of  equal 
amount  in  any  other  set  perpendicular  to  those,  is  the  same  as  a 
simple  shear  in  either  of  the  two  sets  of  planes  cutting  the  two 
sets  of. parallels  at  45°.  And  the  numerical  measure  (§  154)  of  this 
shear,  or  simple  distortion,  is  equal  to  double  the  amount  of  the 
elongation  or  contraction  (each  measured,  of  course,  per  unit  of 
length).  Similarly,  we  see  (§  639)  that  a  longitudinal  traction  (or 
negative  pressure)  parallel  to  one  line,  and  an  equal  longitudinal 
positive  pressure  parallel  to  any  line  at  right  angles  to  it,  is  equivalent 
to  a  distorting  stress  of  tangential  tractions  (§  632)  parallel  to  the 
planes  which  cut  those  lines  at  45°.  And  the  numerical  measure 
of  this  distorting  stress,  being  (§  633)  the  amount  of  the  tangential 
traction  in  either  set  of  planes,  is  equal  to  the  amount  of  the  positive 
or  negative  normal  pressure,  not  doubled. 

653.  Since  then  any  stress  v.-hatever  may  be  made  up  of  simple 
longitudinal  stresses,  it  follows  that,  to  find  the  relation  between  any 
stress  and  the  strain   produced   by  |  iP 

it,  we  have  only  to  find  the  strain  |  ,„ 

produced  by  a  single  longitudinal  |  " 

stress,     which     we     may     do     at  A 

once  thus: — A  simple  longitudinal 
stress,  F,  is  equivalent  to  a  uni- 
form dilating  tension  \F  in  all 
directions,  compounded  with  two  \P  iP^ 
distorting  stresses,  each  equal  to 
~  F,  and  having  a  common  axis 
in  the  line  of  the  given  longitu- 
dinal stress,  and  their  other  two 
axes  any  two  lines  at  right  angles 
to  one  another  and  to  it.  The 
diagram,  drawn  in  a  plane  through 
one  of  these  latter  Unes,  and  the 
former,  sufficiently  indicates  the  synthesis;  the  only  forces  not  shown 
being  those  perpendicular  to  its  plane. 


346  ABSTRACT  DYNAMICS. 

Hencfe  if  n  denote  the  rigidity,  and  k  the  resistance  to  dilatation 
[being  the  same  as  the  reciprocal  of  the  compressibility  (§  651)],  the 
effect  will  be  an  equal  dilatation  in  all  .directions,  amounting,  per 
unit  of  volume,  to 

1  P 

T  <■) 

compounded  with  two  equal  distortions,  each  amounting  to 

(2) 


1  P 


and  having  (§  650)  their  axes  in  the  directions  just  stated  as  those 
of  the  distorting  stresses. 

654.  The  dilatation  and  two  shears  thus  determined  may  be 
conveniently  reduced  to  simple  longitudinal  strains  by  still  following 
the  indications  of  §  652,  thus: — 

The   two  shears  together  constitute  an  elongation  amounting  to 

IP  . 

- —  in  the  direction  of  the  given  force,  P,  and  equal   contraction 

1  p 

amounting  to  - —  in  all  directions  perpendicular  to  it.      And  the 

}  P .       . 

cubic  dilatation  2_  implies  a  lineal  dilatation,  equal  in  all  directions, 


amounting  to  ^-r- .     On  the  whole,  therefore,  we  have 

linear  elongation  =P\  —  +  — 7  j ,  in  the  direction  of  the  applied 
stress,  and 

linear  contraction =/'(-: 7  j,  in  all  directions  perpendicular 

to  the  applied  stress. 


1 
•    (3) 


655.  Hence  when  the  ends  of  a  column,  bar,  or  wire,  of  isotropic 
material,  are  acted  on  by  equal  and  opposite  forces,  it  experiences 

•7  ^ 2  // 

a  lateral  lineal  contraction,  equal   10   -y^, r  of  the  longitudinal 

2(3^  +  n) 

dilatation,  each  reckoned  as  usual  per  unit  of  lineal  measure.  One 
specimen  of  the  fallacious  mathematics  above  referred  to  (§  644),  is 
a  celebrated  conclusion  of  Navier's  and  Poisson's  that  this  ratio 
is  \,  vvhich  requires  the  rigidity  to  be  \  of  the  resistance  to  com- 
pression, for  all  solids :  and  which  was  first  shown  to  be  false  by 
Stokes'  from  many  obvious  observations,  proving  enormous  discre- 
pancies from  it  in  many  well-known  bodies,  and  rendering  it  most 
improbable*^that  there  is  any  approach  to  a  constancy  of  ratio  between 

'  '.,0n  the  Friction  of  Fluids  in  Motion,  and  the  Equilibrium  and  Motion  of 
Elastic  Solids  '  'Trans.  Camb.  Phil.  Sec,  April  1845.  See  also  Camb.  and 
Dub:  Math.  Jour.,  March  18+8. 


STATICS   OF  SOLIDS   AND   FLUIDS  247 

rigidity  and  re<^istince  to  compre-s'on  in  any  class  of  solids.  Thus 
clear  elastic  jellies,  and  India  rubber,  present  familiar  specimens  of 
isotropic  homogeneous  solids,  which,  while  differing  very  much  from 
one  another  in  rigidity  ('stiffness'),  are  probably  all  of  very  nearly 
the  same  compressibility  as  water.  This  being  •g^j^'.Txnj  per  pound 
per  square  inch;  the  resistance  to  compression,  measured  by  its 
reciprocal,  or,  as  we  may  read  it,  '308000  lbs.  per  square  inch,' 
is  obviously  many  hundred  times  the  absolute  amount  of  the 
rigidity  of  the  stiffest  of  those  substances.  A  column  of  any  of 
them,  therefore,  when  pressed  together  or  pulled  out,  within  its  limits 
of  elasticity,  by  balancing  forces  applied  to  its  ends  (or  an  India 
rubber  band  when  pulled  out),  experiences  no  sensible  change  of 
volume,  though  a  very  sensible  change  of  length.  Hence  the  pro- 
portionate extension  or  contraction  of  any  transverse  diameter  must 
be  sensibly  equal  to  ^  the  longitudinal  contraction  or  extension:  and 
for  all  ordinary  stresses,  such  substances  may  be  practically  regarded 
as  incompressible  elastic  solids.  Stokes  gave  reasons  for  believing 
that  metals  also  have  in  general  greater  resistance  to  com.pression,  in 
proportion  to  their  rigidities,  than  according  to  the  fallacious  theory, 
although  for  them  the  discrepancy  is  ver)'  much  less  than  for  the 
gelatinous  bodies.  This  probable  conclusion  was  soon  experimentally 
demonstrated  by  Wertheim,  who  found  the  ratio  qf  lateral  to  longi- 
tudinal change  of  lineal  dimensions,  in  columns  acted  on  solely  by 
longitudinal  force,  to  be  about  \  for  glass. or  brass;  and  by  Kirchhoff, 
who,  by  a  very  well-devised  experimental  method,  found  •3S7  as  the 
value  of  that  ratio  for  brass,  and  "294  for  iron.  For  copper  we  find 
that  it  probably  lies  between  '226  and  '441,  by  recent  experiments' 
of  our  own,  measuring  the  torsional  and  longitudinal  rigidities  (§§  609, 
657)  of  a  copper  wire. 

656.  All  these  results  indicate  rigidity  kss  in  proportion  to  the 
compressibility  than  according  to  Navier's  and  Poisson's  theory. 
And  it  has  been  supposed  by  many  naturalists,  v,ho  have  seen  the 
necessity  of  abandoning  that  theory  as  inapplicable  to  ordinary  solids, 
that  it  may  be  regarded  as  the  proper  theory  for  an  ideal  perfect  so/iJ, 
and  as  indicating  an  amount  of  rigidity  not  quite  reached  in  any 
real  substance,  but  approached  to  in  some  of  the  most  rigid  of 
natural  solids  (as,  for  instance,  iron).  But  it  is  scarcely  possible 
to  hold  a  piece  of  cork  in  the  hand  without  perceiving  the  fallacious- 
ness of  this  last  attempt  to  mamtain  a  theory  which  never  had  any 
good  foundation.  By  careful  measurements  on  columns  of  cork 
of  various  forms  (among  them,  cylindrical  pieces  cut  in  the  ordmary 
way  for  bottles)  before  and  after  compressing  them  longitudinally  in  a 
Brahmah's  press,  we  have  found  that  the  change  of  lateral  dimensions 
is  insensible  both  with  small  longitudinal  contractions  and  return 
dilatations,  within  the  limits  of  elasticity,  and  with  such  enormous 
longitudinal  contractions  as  to  ^  or  I  of  the  original  length.  It  is 
thus  proved  decisively  that  cork  is  much  more  rigid,  .while  metals-, 

*  *  On  the  Elasticity  and  Viscosity  of  Metals'  (VV,  Thoiiiion),  F/::.  I'.S.,  .'lay 
1860 


248  ABSTRACT  DYNAMICS. 

glass,  and  gelatinous  bodies  are  all  less  rigid,  in  proportion  to 
resistance  to  compression,  than  the  supposed  'perfect  solid';  and  the 
utter  worthlessness  of  the  theory  is  experimentally  demonstrated. 

657.  The  modulus  of  elasticity  of  a  bar,  wire,  fibre,  thin  filament, 
band,  or  cord  of  any  material  (of  which  the  substance  need  not  be 
isotropic,  nor  even  homogeneous  within  one  normal  section),  as  a 
bar  of  glass  or  wood,  a  metal  wire,  a  natural  fibre,  an  India  rubber 
band,  or  a  common  thread,  cord,  or  tape,  is  a  term  introduced 
by  Dr.  Thomas  Young  to  designate  what  we  also  sometimes  call  its 
longitudinal  rigidity  that  is,  the  quotient  obtained  by  dividing  the 
simple  longitudinal  force  required  to  produce  any  infinitesimal 
elongation  or  contraction  by  the  amount  of  this  elongation  or  con- 
traction reckoned  as  always  per  unit  of  length. 

658.  Instead  of  reckoning  the  modulus  in  units  of  weight,  it  is 
sometimes  convenient  to  express  it  m  terms  of  the  weight  of  the  unit 
length  of  the  rod,  wire,  or  thread.  The  modulus  thus  reckoned, 
or,  as  it  is  called  by  some  writers,  the  length  of  the  modulus,  is 
of  course  found  by  dividing  the  weight-modulus  by  the  weight  of 
the  unit  length.  It  is  useful  in  many  applications  of  the  theory  of 
elasticity;  as,  for  instance,  in  this  result,  which  will  be  proved 
later- — the  velocity  of  transmission  of  longitudinal  vibrations  (as  of 
sound)  along  a  bar  or  cord,  is  equal  to  the  velocity  acquired  by 
a  body  m  falhng  from  a  height  equal  to  half  the  length  of  the 
modulus' 

659.  The  specific  modulus  of  elasticiiy  of  an  isotropic  substance,  or, 
as  It  is  most  often  called,  simply  the  modulus  of  elasticity  of  the  sub- 
stance, IS  the  modulus  of  elasticity  of  a  bar  of  it  having  some  definitely 
specified  sectional  area.  If  this  be  such  that  the  weight  of  unit 
length  is  unity,  the  modulus  of  the  substance  will  be  the  same  as  the 
length  of  the  modulus  of  any  bar  of  it;  a  system  of  reckoning  v\fhich, 
as  v/e  have  seen,  has  some  advantages  in  application.  It  is,  how- 
ever, more  usual  to  choose  a  common  unit  of  area  as  the  sectional 
area  of  the  bar  referred  to  in  the  definition.  There  must  also  be  a 
definite  understanding  as  to  the  unit  in  terms  of  which  the  force  is 
measured,  which  may  be  either  the  absolute  unit  (§  i88):  or  the 
gravitation  unit  for  a  specified  locality;  that  is  (§  191),  the  weight  in 
that  locality  of  the  unit  of  mass.  Experimenters  hitherto  have  stated 
their  results  in  terms  of  the  gravitation  unit,  each  for  his  own  locality; 
the  accuracy  hitherto  attained  being  scarcely  in  any  cases  sufficient  to 

'  It  IS  to  be  understood  that  the  vibrations  in  question  are  so  much  spread  out 
through  the  leiigth  of  the  body,  that  inertia  does  not  sensibly  influence  the  trans- 
verse contractions  and  dilatations  which  (unless  the  substance  have  in  this  respect 
the  peculiar  character  presented  by  cork,  §  656)  take  place  along  with  them.  Also, 
under  Thermodynamics,  we  shall  see  that  changes  of  temperature  produced  by  the 
varying  strains  cause  changes  of  stress  which,  in  ordinary  solids,  render  the  velocity 
of  transmission  of  longitudinal  vibrations  sensibly  greater  than  that  calculated  by 
the  rule  stated  in  the  text,  if  we  use  the  static  modtdus  ab  understood  from  the 
definition  there  given;  and  we  shall  learn  to  take  into  account  the  thermal  effect 
by  using  a  definite  static  modulus,  or  kiiutic  modulus,  according  to  the  circumstances 
of  any  case  that  may  occur. 


STATICS   OF  SOLIDS  AND   FLUIDS.  249 

require  corrections  for  the  different  forces  of  gravity  in  the  different 
places  of  observation. 

660.  The  most  useful  and  generally  convenient  specification  of 
the  modulus  of  elasticity  of  a  substance  is  in  grammes-weight  per 
square  centimetre.  This  has  only  to  be  divided  by  the  specific 
gravity  of  the  substance  to  give  the  It^rigth  0/  the  modulus.  British 
measures,  however,  being  still  unhappily  sometimes  uSed  in  practical 
and  even  in  high  scientific  statements,  we  may  have  occasion  to  refer 
to  reckonings  of  the  modulus  in  pounds  per  square  inch  or  per  square 
foot,  or  to  length  of  the  modulus  in  feet. 

661.  The  reckoning  most  commonly  adopted  in  British  treatises 
on  mechanics  and  practical  .statements  is  pounds  per  square  mch. 
The  modulus  thus  stated  must  be  divided  by  the  weight  of  12  cubic 
mches  of  the  solid,  or  by  the  product  of  its  specific  gravity  into  '4337', 
to  find  the  length  of  the  modulus,  in  feet. 

To  reduce  from  pounds  per  square  inch  to  grammes  per  square 
centimetre,  multiply  by  70"3i,  or  divide  by  •014223.  French  engineers 
generally  state  their  results  in  kilogrammes  per  square  millimetre, 
and  so  bring  them  to  more  convenient  numbers,  being  -^(s (}(]()»  of  ^^he 
inconveniently  large  numbers  expressing  moduli  in  grammes-weight 
per  square  centimetre. 

662.  The  same  statements  as  to  units,  reducing  factors,  and  nominal 
designations,  are  applicable  to  the  resistance  to  compression  of  any 
elastic  solid  or  fluid,  and  to  the  rigidity  (§  651)  of  an  isotropic  body; 
or,  m  general,  to  any  one  of  the  21  co-efficients  in  the  expressions 
for  terms  in  stresses  of  strains,  or  to  the  reciprocal  of  any  one  of 
the  21  co-efficients  in  the  expressions  for  strains  in  terms  of  stresses, 
as  well  as  to  the  modulus  defined  by  Young. 

663.  In  §§  65  2,  653  we  examined  the  effect  of  a  simple  longitudinal 
stress,  in  producing  elongation  in  its  own  direction,  and  contraction 

'  This  decimal  being  the  weight  In  pounds  of  11  cubic  inches  of  water.  The 
one  great  advantage  of  the  French  metrical  system  is,  that  the  mass  of  the  unit 
volume  (1  cubic  centimetre)  of  water  at  its  tempeiature  of  maximum  density 
<  5'-"94  5  C.)  is  unity  (i  gramme)  to  a  sufficient  degree  of  approximation  for  almost 
ail  practical  purposes.  Thus,  according  to  this  system,  the  density  of  a  body  and 
Its  specific  gravity  mean  one  and  the  same  thing;  whereas  on  the  British  no-system 
the  density  is  expressed  by  a  number  found  by  multiplying  the  specific  gravity  by 
one  number  or  another,  according  to  the  choice  of  a  cubic  inch,  cubic  foot,  cubic 
yard,  or  cubic  mile  that  is  made  for  the  unit  of  volume,  and  the  grain,  scruple, 
gunmaker's  drachm,  apothecary's  drachm,  ounce  Troy,  ounce  avoirdupois,  pound 
Troy,  pound  avoirdupois,  stone  (Imperial,  Ayrsnire,  Lanarkshire,  Dumbarton- 
shire), stone  for  hay,  stone  for  com,  quarter  (of  a  hundredweight),  quarter  (of 
corn),  hundredweight,  or  ton,  that  is  chosen  for  unit  of  mass.  It  is  a  remarkable 
phenomenon,  belonging  rather  to  moral  and  social  than  to  physical  science,  that 
a  people  tending  naturally  to  be  regulated  by  common  sense  should  voluntarily 
condemn  themselves,  as  the  British  have  so  long  done,  to  unnecessary  hard  labour 
in  every  action  of  common  business  or  scientific  work  related  to  measurement,  from 
which  all  the  other  nations  of  Europe  have  emancipated  themselves.  We  have 
been  informed,  through  the  kindness  of  Professor  W.  H.  Miller,  of  Cambridge, 
that  he  concludes,  from  a  very  tiuitwoithy  comparisoa  of  standirds  by  KupHer,  of 
St.  Petersburgh,  that  the  weight  of  a  cubic  decizuctie  of  water  at  teiiif>erature  of 
rnxxixnum  density  is  looo'oij  giaiajiiis. 


250 


ABSTRACT  DYNAMICS. 


in  lines  perpendicular  to  it.  With  stresses  substituted  for  strains,  and 
strains  for  stresses,  we  may  apply  the  same  process  to  mvestigate  the 
longitudinal  and  lateral  tractions  required  to  produce  a  simple  longi- 
tudinal strain  (that  is,  an  elongation  in  one  direction,  with  no  change 
of  dimensions  perpendicular  to  it)  m  a  rod  or  solid  of  any  shape. 

Thus  a  pimple  longitudinal  strain  e  is  equivalent  to  a  cubic  dilata- 
tion e  without  change  of  figure  (or  linear  dilatation  \e  equal  in  all 
directions),  and  two  distortions  consisring  each  of  dilatation  \e  in  the 
given  direction,  and  contraction  \e  in  each  of  two  directions  perpen- 
dicular to  it  and  to  one  another.  To  produce  the  cubic  dilatation,  e, 
alone  requires  (§  651)  a  normal  traction  ke  equal  in  all  directions. 
And,  to  produce  either  of  the  distortions  simply,  sjnce  the  measure 
(§  154)  of  each  is  \e,  requires  a  distorting  stress  equal  to  «  x  |^,  which 
consists  of  tangential  tractions  each  equal  to  this  amount,  positive  (or 
drawing  outwards)  in  the  line  of  the  given  elongation,  and  negative  (or 
pressing  inwards)  in  the  perpendicular  direction.     Thus  we  have  in  all 

normal  traction  =  (/^  +  ^«)  <?,  in  the  direction  of  the  given] 
strain,  and  | 


normal  traction  =(>^-§«)  ^,  in   evehy  direction  perpen- 1 
dicular  to  the  given  strain.  I 


(4) 


664.  If  now  we  suppose  any  possible  infinitely  small  strain  {e,f,  g, 
a,  by  c),  according  to  the  specification  of  §  640,  to  be  given  to  a  body, 
the  stress  {F,  Q,  R,  S,  T,  U)  required  to  maintain  it  will  be  expressed 
by  the  following  formulae,  obtained  by  successive  applications  of 
§  663  (4)  to  the  components  e,  f,  g  separately?  and  of  §  651  to 
a,  b,  c: — 

S=  na,  T=  nb,  U=  nc,  T 

/^=2l^+S9(^+/), 


wher? 


%  =  k  + 


4« 


■1 


C»> 


665.    Similarly,  by  §  651  and  §  654  (3),  we  have 


a  =  -S,b. 
n 


where 
and 


Zc^-'-U, 


Me^{P-<j{Q^R))^ 
M/={Q-cr{R  +  P)], 

()nk 


M  = 


Zk  +  n 
2  (3> 


M 


(6) 


t+«)~^^"^'    J 


STATICS    OF  SOLIDS   AXD    FLUIDS.  251 

as  the  formulae  expressing  the  strain  (1^,  /,  g,  a.  A,  l)  in  terms  of 
the  stress  {P,  Q,  R,  S,  T,  if).  i'hcy  aie  ot  coarse  merely  the 
algebraic  inversions  of  (5),  and  they  iiughc  have  been  found  by 
solvmg  these  for  e,/,g,  a,  b,  c,  ret^arded  as  the  unknown  quantities. 
Mis  here  introduced  to  denote  Young's  modulus. 

666.  To  express  the  equation  ol'  energy  tor  an  isotropic  substance, 
we  may  take  the  general  formula, 

7£/  =  i  (/V  +  Q/+  Pg  ~  Sa  +  Tl)  +  C:-), 
and  eliminate  from  it  P,  Q,  etc.,  by  (5)  of  ^  0O4,  or,  again,  e,  /,  etc., 
by  (6)  of  §  665,  we  thus  find 

667.  The  mathematical  theory  of  the  equilibrium  of  an  elastic 
solid  presents  the  following  general  problerhs  : 

A  solid  cf  any  given  s/iapt:,  when  undiiturled,  is  acted  on  in  its 
substance  by  force  distributed  through  it  in  arty  given  manner,  and  dis- 
placements are  arbitrarily  produced,  or  forces  arbitrarily  applied,  oxer 
its  bounding  surface.  It  is  required  to  find  the  displacement  of  every 
point  of  its  substance. 

This  problem  has  been  thoroughly  solved  for  a  shell  of  homo- 
geneous isotropic  substance  bounded  by  surfaces  which,  when  undis- 
turbed, are  spherical  and  concentric ;  but  not  hitherto  for  a  body 
of  any  other  shape.  The  limitations  under  which  solutions  have 
been  obtained  for  other  cases  (thin  plates  and  rods),  leading,  as  we 
have  seen,  to  important  practical  results,  have  been  stated  above 
(§  605).  To  demonstrate  the  laws  (§  607)  which  were  taken  m 
anticipation  will  also  be  one  of  our  applications  of  the  general 
equations  for  interior  equilibrium  of  an  elastic  solid,  which  we  now 
proceed  to  investigate. 

668.  Any  portion  in  the  interior  of  an  elastic  solid  may  be 
regarded  as  becoming  perfectly  rigid  (.^  584)  without  disturbing  the 
equilibrium  either  of  itself  or  of  the  matter  round  it.  Hence  the  traction 
exerted  by  the  matter  all  round  it,  regarded  as  a  distribution  of  force 
applied  to  its  surface,  must,  with  the  applied  forces  acting  on  the  sub- 
stance of  the  portion  considered,  fulfil  the  conditions  of  equihbrium  of 
forces  acting  on  a  rigid  body.  This  statem.ent,  applied  to  an  infinitely 
small  rectangular  parallelepiped  of  the  body,  gives  the  general  differ- 
ential equations  of  internal  equilibrium  of  an  elastic  solid.  It  is  to  be 
remarked  that  three  equations  suflSce ;  the  conditions  of  equilibrium 
for  the  couples  being  secured  by  the  relation  established  above  (§632) 
among  the  six  pairs  of  tangential  component  tractions  on  the  six 
faces  of  the  figure. 

669.  One  of  the  most  beautiful  applications  of  the  general  equa- 
tions  of  internal   equilibrium  of  an  elastic  solid  hitherto  made  is 


252  ABSTRACT  DYNAMICS. 

that  of  M.  de  St.  Venant  to  'the  torsion  of  prisms'.'  To  one 
end  of  a  long  straight  prismatic  rod,  wire,  or  solid  or  hollow  cylinder 
of  any  form,  a  given  couple  is  applied  in  a  plane  perpendicular  to 
the  le'ngth,  while  the  other  end  is  held  fast :  it  is  required  to  find 
the  degree  of  twist  produced,  and  the  distribution  of  strain  and 
stress  throughout  the  prism.  The  conditions  to  be  satisfied  here 
are  that  the  resultant  action  between  the  substance  on  the  two  sides 
of  any  normal  section  is  a  couple  in  the  normal  plane,  equal  to  the 
given  couple.  Our  work  for  solving  the  problem  will  be  much 
simplified  by  first  establishing  the  following  preliramary  propo- 
sitions : — 

670.  Let  a  solid  (whether  aeolotropic  or  isotropic)  be  so  acted 
on  by  force  applied  from  without  to  its  boundary,  that  throughout  its 
interior  there  is  no  normal  traction  on  any  plane  parallel  or  per- 
pendicular to  a  given  plane,  XO  V,  which  implies,  of  course,  that 
there  is  no  distorting  stress  with  axes  in  or  parallel  to  this  plane,  and 
that  the  whole  stress  at  any  point  of  the  solid  is  a  simple  distorting 
stress  of  tangential  forces  in  some  direction  in  the  plane  parallel  to 
XO  V,  and  in  the  plane  perpendicular  to  this  direction.    Then — 

(i)  The  interior  distorting  stress  must  be  equal,  and  similarly 
directed,  in  all  parts  ot  the  solid  lying  in  any  line  perpendicular 
to  the  plane  XO  Y. 

(2)  It  being  premised  that  the  traction  at  every  point  of  any 
surface  perpendicular  to  the  plane  XOY  is,  by  hypothesis,  a  distribu- 
tion offeree  in  lines  perpendicular  to  this  plane  ;  the  integral  amount 
of  it  on  any  closed  prismatic  or  cylindrical  surface  perpendicular  to 
XO  Y,  and  bounded  by  planes  parallel  to  it,  is  zero. 

(3)  The  matter  within  the  prismatic  surface  and  terminal  planes  of, 
(2)  being  supposed  for  a  moment  (§  584)  to  be  rigid,  the  distribution 

of  tractions  referred  to  in 
(2)  constitutes  a  couple 
whose  moment,  divided  by 
the  distance  between  those 
terminal  planes,  is  equal  to 
the  resultant  force  of  the 
tractions  on  the  area  of 
either,  and  whose  plane  is 
parallel  to  the  lines  of  these 
resultant  forces.  In  other 
words,  the  moment  of  the 
qT  '~  Ix"  distribution  of  forces  over 

the  prismatic  surface  referred  to  in  (2)  round  any  line  (OY or  OX)  in 
the  plane  XOY,  is  equal  to  the  sum  of  the  components  (7"  or  S), 
perpendicular  to  the  same  line,  of  the  traction  in  eitlier  of  the 
terminal  planes  multiplied  by  the  distance  between  these  planes. 

'  Menwires  tUs  Savants  Eiran^ns.  1855.  '  Do  la  Torsion  des  Pnsmes,  avec  des 
ConsidenitioQE  sui  leur  Flexiou,  etc. 


STATICS  OF  SOLIDS  AND  FLUIDS  253 

To  prove  (i)  consider  for  a  moment  as  rigid  (§  584)  an  infinite- 
simal  prisnl,  AB   (of  sectional   area  w),   per-        '  . 

pendicular  to  XO  Y,  and  having   plane  ends,  m >  Tu 

A,  £,  parallel  to  it.     There  being  no  forces  j  j 

on  iis  sides  (or  cylindrical  boundary)  per- 
pendicular to  its  length,  its  equilibrium  so  far 
as  motion  m  the  direction  of  any  line  {OX), 
perpendicular  to  its  length,  requires  that  the 
components  of  the  tractions  on  its  ends  be 
equal  and  in  opposite  directions.  Hence, 
in  the  notation  §  633,  the  distorting- stress 
components,  T,  must  be  equal  at  A  and  £ ; 

and  so  must  the  stress  components  S,  for  the    ^^"^ ^ 

same  reason. 

To  prove  (2)  and  (3)  we  have  only  to  remark  that  they  are 
required  for  the  equilibrium  of  the  rigid  prism  referred  to 
in  (3). 

671.  For  a  solid  or  hollow  circular  cylinder,  the  solution  of  §  669 
(given  first,  we  believe,  by  Coulomb)  obviously  is  that  each  circular 
normal  section  remains  unchanged  in  its  own  dimensions,  figure, 
and  internal  arrangement  (so  that  ever)-  straight  line  of  its  particles 
remains  a  straight  line  of  unchanged  length),  but  is  turned  round 
the  axis  of  the  cylinder  through  such  an  angle  as  to  give  a  uniform 
rate  of  haist  equal  to  the  applied  couple  divided  by  the  product 
of  the  moment  of  inertia-  of  the  circular  area  (whether  annular  or 
complete  to  the  centre)  into  the  rigidity  of  the  substance. 

672.  Similarly,  we  see  that  if  a  cylinder  or  prism  of  any  shape 
be  compelled  to  take  exactly  the  state  of  strain  above  specified  (§  671) 
with  the  line  through  the  centres  of  inertia  of  the  normal  sections, 
taken  instead  of  the  axis  of  the  cylinder,  the  mutual  action  between 
the  parts  of  it  on  the  two  sides  of  any  normal  section  will  be  a  couple 
of  which  the  moment  will  be  expressed  by  the  same  formula,  that  is, 
the  product  of  the  rigidity,  into  the  rate  of  twist,  into  the  niomeiu 
of  inertia  of  the  section  round  its  centre  of  inertia. 

673.  But  for  any  other  shape 'of  prism  than  a  solid  or  symm.etncal 
hollow  circular  c)!inder,  the  supposed  state  of  strain  v.-ill  require, 
besides  the  terminal  opposed  couples,  force  parallel  to  the  length 
of  the  prism,  distributed  over  the  prismatic  boundary,  m  proportion 
to  the  distance  along  the  tangent,  from  each  point  of  the  surface, 
to  the  point  in  which  this  line  is  cut  by  a  perpendicular  to  it  from  the 
centre  of  inertia  of  the  normal  section.  To  prove  this  let  a  normal 
section  of  the  prism  be  represented  in  the  annexed  diagram  (i^age  254). 
Let  PK  representing  the  shear  at  any, point,  P,  close  to  the  prismatic 
boundary,  be  resolved  into  PN  and  VT  respectively  along  the  nor- 
mal and  tangent.  The  whole  shear,  PK,  being  equal  to  t;-,  us 
component, /'yV,  is  equal  to  rr  sin  01  or  t./'/T.  The  corresponding 
component  of  the  required  stress  is  vr.PE,  and  involves  (,5  632) 
equal  forces  in  the  plane  of  the  diagram,  and  in  the  plane  through 


254 


ABSTRACT  DYNAMICS. 


TP  perpendicular  to  it,   each  amounting  to  m.PE  per  unit  of 
area. 

An  application  of  force 
equal  and  opposite  to  the 
distribution  thus  found 
over  the  prismatic  boun- 
dary, would  of  course  alone 
produce  in  the  prism,  other- 
wise free,  a  state  of  stram 
which,  compounded  with 
that  supposed  above,would 
give  the  state  of  strain  ac- 
tually produced  by  the  sole 
application  of  balancing 
couples  to  the  two  ends. 
The  tesult,  it  is  easily  seen  (and  it  will  be  proved  below),  consists  of 
an  increased  twist,  together  with  a  warping  of  naturally  plane  normal 
sections,  by  infmitesimal  displacements  perpendicular  to  themselves, 
into  certain  surfaces  of  anticlastic  curvature,  with  equal  opposite 
curvatures  in  the  principal  sections  (§  122)  through  every  point. 
This  theory  is  due  to  St.  Venant,  who  not  only  pointed  out  the  falsity 
of  the  supposition  admitted  by  several  previous  writers,  that  Cou- 
lomb's law  holds  for  other  forms  of  prism  than  the  solid  or  hollow 
circular  cyhnder,  but  discovered  fully  the  nature  of  the  requisite 
correction,  reduced  the  determination  of  it  to  a  problem  of  pure 
mathematics,  worked  out  the  solution  for  a  great  variety  of  important 
and  curious  cases,  compared  the  results  with  observation  in  a  manner 
satisfactory  and  interesting  to  the  naturalist,  and  gave  conclusions 
of  great  value  to  the  practical  engineer. 

674.  We  take  advantage  of  the  identity  of  mathematical  conditions 
in  St.  -Venant's  torsion  problem,  and  a  hydrokinetic  problem  first 
solved  a  few  years  earlier  by  Stokes ',  to  give  the  following  statement, 
which  will  be  found  very  useful  in  estimating  deficiencies  in  torsional 
rigidity  below  the  amount  calculated  from  the  fallacious  extension 
■of  Coulomb's  law  : — 

675.  Conceive  a  liquid  of  density  n  completely  filling  a  closed 
infinitely  light  prismatic  box  of  the  same  shape  within  as  the  given 
elastic  pnsm  and  of  length  unity,  and  let  a  couple  be  applied  to  the 
box  in  a  plane  perpendicular  to  its  length.  The  effective  moment 
of  inertia  of  the  liquid*  will  be  equal  to  the  correction  by  which  the 
torsional  rigidity  of  the  elastic  prism  calculated  by  the  false  extension 
of  Coulomb's  law,  must  be  diminished  to  give  the  true  torsional 
rigidity 

Further,  the  actual  shear  of  the  solid,  in  any  infinitely  thin  plate  of 

^  'On  some  cases  of  Fluid  Motion,'  Cambridge  Philosophkal  Transactions,  1843. 

'  That  is  the  moment  of  inertia  of  a  rigid  solid  which,  as  will  be  proved  ih 
Vol.  II.,  may  be  fixed  within  the  box,  if  the  liquid  be  removed,  to  make  its 
motions  the  same  as  they  are  with  the  liquid  in  it. 


STATICS   OF  SOL  IDS  AND   FLUIDS. 


255 


it  between  two  normal  sections,  will  at  each  point  be,  when  reckoned 
as  a  differential  sliding  (§  151)  parallel  to  their  planes,  equal  to  and 
in  the  same  direction  as  the  velocity  of  the  liquid  relatively  to  the 
containing  box. 


67.6.  St.  Venant's  treatise  abounds  in  beautiful  and  instructive 
graphical  illustrations  of  his  results,  from  which  we  select  the 
following  : — 

(i)  Elliptic  cylinder.  The  plain  and  dotted  curvilineal  arcs  are 
'contour  lines '  {coupes  topo^raphiques)  of  the  section  as  warped  by 


torsion ;  that  is  to  say,  lines  in  which  it  is  cut  by  a  series  of  parallel 
planes,  each  perpendicular  to  the  axis.  These  lines  are  equilateral 
hy^Derbolas  in  this  case.  The  arrows  indicate  the  direction  of  rotation 
in  the  part  of  the  prism  above  the  plane  of  the  diagram. 

(2)     Eqicilateral  triangular  prism. — The  contour  lines  are  shown 


as  in  case  (i);  the   dotted   curves  being   those  where  the  warped 
section  falls  below  the  plane  of  the  diagram,  the  direction  of  rotation 


256 


ABSTRACT  DYNAMICS. 


of  the  part  of  the  prism .  above  the  plane  being  indicated  by  the 
bent  arrow. 

(3)  This  diagram  shows  a  series  of  lines  given  by  St.  Venant, 
and  more  or  less  resembling'  squares,  their  common  equation 
containing  only  one  constant  a.      It  is  remarkable  that  the  values 


<?  =  o-5  .and  a  =  -\(j2-\)  give  similar  but  not  equal  curvi- 
lineal  squares  (hollow  sides  and  acute  angles),  one  of  them  turned 
through  half  a  right  angle  relatively  to  the  other.  Everything  in  the 
diagram  outside  the  larger  of  _these  squares  i^  to  be  cut  away  as 
irrelevant  to  the  physical  problem;  the  series  of  closed  curves 
remaining  exhibits  figures  of  prisms,  for  any  one  of  which  the  torsion 
problem  is  solved  algebraically.  These  figures  vary  continuously  from 
a  circle,  inwards  to  one  of  the  acute-angled  squares,  and  outwards  to 
the  other :  each,  except  these  extremes,  being  a  continuous  closed 
curve  with  no  angles.  The  curves  for  a  =  0-4  and  a-  — 0-2  approach 
remarkably  near  to  the  rectilineal  squares,  partially  indicated  in  the 
diagram  by  dotted  lines, 

(4)  This  diagram  shows  the  contour  lines,  in  all  respects  as  in  the 
cases  (i)  and  (2)  for  the  case  of  a  prism  having  for  section  the  figure 
indicated.  The  portions  of  curve  outside  the  continuous  closed  curve 
are  merely  indications  of  mathematical  extensions  irrelevant  to  the 
physical  problem. 


STATICS  OF  SOLIDS  AND  fWlDS.  257 

Y 


(5)    This  shows,  as  in  the  other  cases,  the  contour  lines  for  the 
'.varped  section  of  a  square  prism  under  torsion. 


258 


ABSTRACT  DYNAMICS. 


^6),  (7),  (8).      These   are   shaded  drawings,  showing  the  appear- 
ances presented  by  elliptic,  square,  and  flat  rectangular  bars  under 


exaggerated  torsion,  as  may  be  realized  with  such  a    substance   as 
India  rubber. 


677.  Inasmuch  as  the  moment  of  inertia  of  a  plane  area  about 
an  axis  through  its  centre  of  inertia  perpendicular  to  its  plane  is 
obviously  equal  to  the  sum  of  its  moments  of  inertia  round  any  two 
axes  through  the  same  point,  at  right  angles  to  one  another  in  its 
plane,  the  fallacious  extension  of  Coulomb's  law,  referred  to  in  §  673, 

would  make  the, torsional  rigidity  of  a  bar  of  any  section  equal  to  -y> 

(§  665)  multiplied  into  the  sum  of  its  flexural  rigidities  (see  below, 
§  679)  in  any  two  planes  at  right  angles  to  one  another  through 
its  length.  The  true  theory,  as  we  have  'seen  (,5  675),  always 
gives  a  torsional  rigidity  less  than  this..  How  great  the  deficiency 
may  be  expected  to  be  in  cases  in  which  the  figure  of  the  section 
presents  projecting  angles,  or  considerable  prominences  (which  may 
be  imagined  from  the  hydrokinetic  analogy  we  have  given  in  §  675), 
has  been  pointed  out  by  M.  de  St.  Venant,  with  the  important 
practical  application,  that  strengthening  ribs,  or  projections  (see,  for 
instance,  the  fourth  annexed  diagram),  such  as  are  introduced  in 
engineering  to  give  stiffness  to  beams,  have  the  reverse  of  a  good 
effect  when  torsional  rigidity  or  strength  is  art  object,  although  they 
are  truly  of  great  value  in  increasing  the  flexural  rigidity,  and  giving 


STATICS  OF  SOLIDS  AND  FLUIDS. 


259 


strength  to  bear  ordinary  strains,  which  are  always  more  or  less 
flexural.  With  remarkable  ingenuity  and  mathematical  skill  he  has 
drawn  beautiful  illustrations  of  this  important  practical  principle  from 
his  algebraic  and  transcendental  solutions.     Thus  for  an  equilateral 


(I) 

Rectilineal 
square. 


0) 
Square  with  curved 
corners  and  hollow 
sides. 


(3) 

Square  with  acute 

angles  and  hollow 

sides. 


(4)     . 
Star  with  four  (5) 

rounded  points,        Equilateral' 
being  a  curve  of        triangle.' 
the  eighth  degree. 


triangle,  and  for  the  rectilineal  and  three  curvilineal  squares  shown 
in  the  annexed  diagram_,  he  finds  for  the  torsional  rigidities  the 
values  stated.  The  number  immediately  below  the  diagram  indicates 
in  each  case  the  fraction  which  the  true  torsional  rigidity  is  of  the 
old  fallacious-  estimate  (§  673)';  the  latter  being  the  product  of  the 
rigidity  of  the  substance  into  the  moment  of  inertia  of  the  cross 
section  round  an  axis  perpendicular  to  its  plane  through  its  centre 
of  inertia.  The  second  number  indicates  in  each  case  the  fraction 
which  the  torsional  rigidity  is  of  that  of  a  solid  circular  cylinder 
of  the  same  sectional  area. 


678.  M.  de  St  Venant  also  calls  attention  to  a  conclusion  from 
his  solutions  which  to  many  may  be  startling,  that  in  his  simpler  cases 
the  places  of  greatest  distortion  are  those  points  of  the  boundaiy 
which  are  nearest  to  the  axis  of  the  twisted  prism  in  each  case,  and 
the  ])laces  of  least  distortion  those  farthest  from  it.  Thus  in  the 
elliptic  cylinder  the  substance  is  most  strained  at  the  ends  of  the  smaller 
principal  diameter,  and  least  at  the  ends  of  the  greater.  In  the 
eciuilateral  triangular  and  square  prisms  there  are  longitudinal  lines  of 
maximum  strain  through  the  middles  of  the  sides.  In  the  oblong 
rectangular  prism  there  are  two  lines  of  greater  maximum  strain 
through  the  middles  of  the  broader  pair  of  sides,  and  two  lines  of  less 
maximum  strain  through  the  middles  of  the  narrow  sides.  The 
strain  is,  as  we  may  judge  from  (§  675)  the  hydrokinetic  analogy, 
excessively  small,  but  not  evanescent,  in  the  projecting  ribs  of  a  prism 
of  the  figure  shown  in  (4)  §  677.  It  is  ([uite  evanescent  infinitely  near 
the  angle,  in  the  triangular  and  rectangular  prisms,  and  in  each  other 
case  as  (3)  of  §  677,  in  which  there  is  a  finite  angle,  whether  acute 
or  obtuse,  projecting  outwards.  This  reminds  us  of  a  general 
remark  we    have    to    make,  although  consideration   of  space    may 


26o  ABSTRACT  DYNAMICS. 

oblige  us  to  leave  it  without  formal  proof.  A  solid  of  any  elastic 
substance,  isotropic  or  aeolotropic,  bounded  by  any  surfaces  pre- 
senting projecting  edges  or  angles,  or  re-entrant  angles  or  edges, 
however  obtuse,  cannot  experience  any  finite  stress  or  strain  in  the 
neighbourhood  oi ^l projecting zsi'^Q  (trihedral,  polyhedral,  or  conical); 
in  the  neighbourhood  of  an  edge,  can  only  experience  simple  longi- 
tudinal stress  parallel  to  the  neighbouring  part  of  the  edge;  and 
generally  experiences  infinite  stress  and  strain  in  the  neighbourhood 
of  a  re-entrant  edge  or  angle;  when  influenced  by  any  distribution 
of  force,  exclusive  of  surface  tractions  infinitely  near  the  angles  or 
edges  in  question.  An  important  application  of  the  last  part  of 
this  statement  is  the  practical  rule,  well  known  in  mechanics,  that 
every  re-entering  edge  or  angle  ought  to  be  rounded  to  prevent  risk 
of  rupture,  in  solid  pieces  designed  to  bear  stress.  An  illustration 
of  these  principles  is  afforded  by  the  complete  mathematical  solution 
of  the  torsion  problem  for  prisms  of  fan-shaped  sections,  such  as 
the  annexed  figures.  In  the  cases  corresponding  to  figures  (4),  (5), 
(6)  below,  the  distortion  at  the  centre  of  the  circle  vanishes  in  (4), 
is  finite  and  determinate  in  (5),  and  infinite  in  (6). 


(I)  (2)  (3) 


679.  Hence  in  a  rod  of  isotropic  substance  the  principal  axes 
of  flexure  (§  609)  coincide  with  the  principal  axes  of  inertia  of  the 
area  of  the  normal  section ;  and  the  corresponding  flexural  rigidities 
are  the  moments  of  inertia  of  this  area  round  these  axes  multi- 
plied by  Young's  modulus.  Analytical  investigation  leads  to  the 
following  results,  due  to  St.  Venant.  Imagine  the  whole  rod  di- 
vided, parallel  to  its  length,  into  infinitesimal  filaments  (prisms  when 
the  rod  is  straight).  Each  of  these  contracts  or  swells  laterally  with 
sensibly  the  same  freedom  as  if  it  were"  separated  from  the  rest  of 
the  substance,  and  becomes  elongated  or  shortened  in  a  straight  line 
to  the  same  extent  as  it  is  really  elongated  or  shortened  in  the  circular 
arc  which  it  becomes  in  the  bent  rod.  The  distortion  of  the  cross 
section  by  which  these  changes  of  lateral  dimensions  are  necessarily 
accompanied  is  illustrated  in  the  annexed  diagram,  in  which  either  the 
whole  normal  section  of  a  rectangular  beam,  or  a  rectangular  area  in 
the  normal  section  of  a  beam  of  any  figure,  is  represented  in  its  strained 
and  unstrained  figures,  v/ith  the  central  point  O  common  to  the  two. 
The  flexure  is  in  i)lancs  perpendicular  to  YOY,,  and  concave  upwards 
(or  towards  X);  G  the  centre  of  curvature,  being  in  the  direction 
indicated,  but  too  far  to  be  included  in  the  diagram.     The  straight 


STATICS   OF  SOLIDS  AND   FLUIDS. 


261 


sides  AC,  BD,  and  all  straight  lines  parallel  to  tTiem,  of  the  unstrained 
rectangular  area  become  concentric  arcs  of  circles  concave  in  the 

\G 


H 


opposite  direction,  their  centre  of  curvature,  H,  being  for  rods  of 
gelatinous  substance,  or  of  glass  or  metal,  from  2  to  4  times  as  far 
frcim  O  on  one  side  as  G  is  on  the  other.  Thus  the  originally  plane 
sides  AC,  BD  of  a  rectangular  bar  become  anticlastic  surfaces,  of 

curvatures  -  and  —  ,  in  the  two  principal  sections.  A  flat  rectans;ular, 

or  a  square,  rod  of  India-rubber  [for  which  u  amounts  (3655)  to  very 
nearly  \,  and  which  is  susceptible  of  very  great  amounts  of  strain 
without  utter  loss  of  corresponding  elastic  action],  exhibits  this 
phenomenon  remarkably  well. 

680.  The  conditional  limitation  (;56o5)  of  the  curvature  to  being  very 
small  in  com[jar;son  with  that  of  a  circle  of  radius  ecjual  to  the  greatest 
diameter  of  the  normal  section  (nut  obviously  necessary,  and  in^leed 
not  generally  known  to  be  necessary,  we  believe^  when  the  greatest 
diameter  is  jjerpendicular  to  tlie  [.lane  of  cur\at*uie)  now  receives  its 
full  explanation.  Fur  unless  \.\w  bnaiitJi,  AC,  of  the  bar  (or  diameter 
perpendicular  to  the  plane  of  lle.xure)  be  very  small  in  comiKiris.on 
with  the  mean  [jroportional  lietween  the  radius,  Oil,  and  the  thiek- 
ne?,s,  AB,  the  cii^tances  fru!n  OV  to  the  corners  y/',  C"  wuuld  fall 
short  of  the  half  thickness,  0£,  and  the  distanees  to  />",  JJ  would 
exceed  it  by  difterenees  comparable  with  its  own  amount.  This 
would  give  rise  to  sensibly  less  and  greater  shortenir.gs  and  stretehmgs 


262  ABSTRACT  DYNAMICS. 

in  the  filaments  towards  the  corners,  and  so  vitiate  the  solution. 
Unhappily  mathematicians  have  not  hitherto  succeeded  in  solving, 
possibly  not  even  tried  to  solve,  the  beautiful  problem  thus  presented 
by  the  flexure  of  a  broad  very  thin  band  (such  as  a  watch  spring)  into 
a  circle  of  radius  comparable  with  a  third  proportional  to  its  thickness 
and  its  breadth. 

681.  But,  provided  the  radius  of  curvature  of  the  flexure  is  not 
only  a  large  multiple  of  the  greatest  diameter,  but  also  of  a  third 
proportional  to  the  diameters  in  and  perpendicular  to  the  plane  of 
flexure;  then  however  great  may  be  the  ratio  of  the  greatest  diameter 
to  the  least,  the  preceding  solution  is  applicable;  and  it  is  remarkable 
that  the  necessary  distortion  of  the  normal  section  (illustrated  in  the 
diagram  of  §  679)  does  not  sensibly  impede  the  free  lateral  con- 
tractions and  expansions  in  the  filaments,  even  in  the  case  of  a  broad 
thin  lamina  (whether  of  precisely  rectangular  section,  or  of  unequal 
thicknesses  in  different  parts). 

682.  In  our  sections  on  hydrostatics,  the  problem  of  finding  the 
deformation  produced  in  a  spheroid  of  incompressible  liquid  by  a 
given  disturbing  force  will  be  solved ;  and  then  we  shall  consider  the 
application  of  the  preceding  methods  to  an  elastic  solid  sphere  in  their 
bearing  on  the  theory  of  the  tides  and  the  rigidity  of  the  earth.  This 
proposed  application,  however,  reminds  us  of  a  general  remark  of 
great  practical  importance,  with  which  we  shall  leave  elastic  solids  for 
the  present.  Considering  different  elastic  solids  of  similar  substance 
and  similar  shapes,  we  see  that  if  by  forces  applied  to  them  in  any 
way  they  are  similarly  strained,  the  surface  tractions  in  or  across 
similarly  situated  elements  of  surface,  whether  of  their  boundaries 
ol:  of  surfaces  imagined  as  cutting  through  their  substances,  must  be 
equal,  reckoned  as  usual  per  unit  of  area.  Hence;  the  force  across, 
or  in,  any  such  surface,  being  resolved  into  components  parallel  to 
any  directions ;  the  whole  amounts  of  each  such  component  for 
similar  surfaces  of  the  different  bodies  are  in  proportion  to  the  squares 
of  their  lineal  dimensions.  Hence,  if  equilibrated  similarly  under  {he 
action  of  gravity,  or  of  their  kinetic  reactions  (§  236)  against  equal 
accelerations  (§  32),  the  greater  body  would  be  more  stramed  than  the 
less;  as  the  amounts  of  gravity  or  of  kinetic  reaction  of  similar 
portions  of  them  are  as  the  ad>cs  of  their  linear  dimensions.  Defi- 
nitively, the  strains  at  similarly  situated  points  of  the  bodies  will 
be  in  simple  proportion  to  their  linear  dimensions,  and  the  displace- 
ments will  be  as  the  squares  of  these  lines,  provided  that  there  is  no 
strain  in  any  part  of  any  of  them  too  great  to  allow  the  principle 
of  superposition  to  hold  with  sufficient  exactness,  and  that  no  part  is 
turned  through  more  than  a  very  small  angle  relatively  to  any  other 
part.  To  illustrate  by  a  single  exajn[)le,  let  us  consider  a  uniform 
long,  thin,  round  rod  held  horizontally  by  its  middle.  Let  its 
substance  be  homogeneous,  of  density  /■>,  and  Young's  modulus,  M; 
and  let  its  length,  /,  be  /  times  its  diameter.  Then  (as  the  moment 
of  inertia  of  a  circular  area  of  radius  r  round  a  diameter  is  \-nr*)  the 


STATICS   OF  SOLIDS  AXD   FLUIDS.  263 

fiexural  rigiuity  of  the  rod  ^vill  (5  679)  be  ^—  tt  ^  -  \  .     This  gives  us 

for  the  curvature  at  the  middle  of  the  rod  the  elongation  and  con- 
traction where  greatest,  that  is,  at  the  highest  and  lowest  points  of  the 
normal  section  through  the  middle  point ;  and  the  droop  of  the  ends; 
the  following  expressions 

M  '    M  '  SJ/ 

Thus,  for  a  rod  whose  length  is  200  times  its  diameter,  if  its  substance 
be  iron  or  steel,  for  whii.ii  p  =  775,  -^-nd  M=  194  x  lo'  granimes  per 
square  centimetre,  the  maximum  elongation  and  contraction  (being 
at  the  top  and  bottom  of  the  middle  section  where  it  is  held)  are 
each  equal  to  S  x  10"*  x  /,  and  the  droop  of  its  ends  2  x  10"^  x  /*. 
Thus  a  steel  or  iron  wire,  ten  centimetres  long,  and  half  a  millimetre 
in  diameter,  held  horizontally  by  its  middle,  would  experience  only 
•oooooS  of  maximum  elongation  and  contraction,  and  only  002  of 
a  centimetre  of  droop  in  its  ends  :  a  round  steel  rod,  of  half  a  centi- 
metre in  diameter,  and  one  metre  long,  would  experienct?  ccooS  of 
maximum  elongation  and  contraction,  and  '2  of  a  centimetre  of 
droop  :  a  round  steel  rod,  of  ten  centimetres  diameter,  and  twenty 
metres  long,  must  be  of  remarkable  temper  (see  Fropcrtics  of  Matter) 
to  bear  being  held  by  the  middle  without  taking  a  wry  sensible  per- 
manent set :  and  it  is  probable  that  no  temper  of  steel  is  high  enough 
in  a  round  shaft  forty  metres  long,  if  only  two  decimetres  in  dia- 
meter, to  allow  it  to  be  held  by  its  middle  without  either  bending 
it  to  some  great  angle,  and  beyond  all  appearance  of  elasticity,  or 
breaking  it. 

683.  In  passing  from  the  dynamics  of  perfectly  elastic  solids  to 
abstract  hydrodynamics,  or  the  dynamics  of  perfect  tluids,  it  is  con- 
venient and  instructive  to  anticipate  slightly  some  of  the  \iews  as  to 
intermediate  properties  observed  in  real  solids  and  tluids,  which, 
according  to  the  general  [jlan  proposed  (§  402)  for  our  work,  will  be 
examined  with  more  detail  under  Fropgrtics  of  Matter. 

By  induction  from  a  great  variety  of  observed  phenomena,  we  are 
compelled  to  conclude  that  no  change  of  volume  or  of  shape  can  be 
produced  in  any  kind  of  matter  without  dissipation  of  energy  (§  247); 
so  that  if  in  any  case  there  is  a  return  to  the  primitive  configuration, 
some  amount  (however  small)  of  work  is  always  required  to  com- 
pensate the  energy  dissipated  away,  and  restore  the  bud)'  to  the  same 
physical  and  the  same  jjalpably  kinetic  condition  as  tiiat  in  which  it 
was  given.  ^Ve  have  seen  (§  643),  by  anticipating  something  of 
thermodynamic  principles,  how  such  dissipation  is  inevitable,  e\en  in 
dealing  with  the  absolutely  perfect  <t\j.<,\.[c\iy  o{\o\umQ  presented  by  every 
fluid,  and  possibly  by  some  solids,  as,  for  instance,  homogeneous 
crystals.  But  in  metals,  glass,  porcelain,  natural  stones,  wood,  India- 
rubber,  homogeneous  jelly,  silk  fibre,  ivory,  etc.,  a  distinct  fnotiona/ 


264  ABSTRACT  DYNAMICS. 

resistance'  against  every  change  of  shape  is,  as  we  shall  see  !ater, 
under  Properties  of  Matter,  demonstrated  by  many  experiments,  and 
is  found  to,  depend  on  the  speed  with  which  the  change  of  shape  is 
made.  A  very  remarkable  and  obvious  proof  of  frictional  resistance 
to"  change  of  shape  in  ordinary  solids,  is  afforded  by  the  gradual, 
more  or  less  rapid,  subsidence  of  vi_brat)ons  of  elastic  solids;  inar- 
vellously  rapid  in  India-rubber,  and  even  in  homogeneous  jelly;  less 
rapid  in  glass  and  metal  springs,  but  still  demonstrably,  much  more 
rapid  than  can  be  accounted  for  by  the  resistance  of  the  air.  This 
molecular  friction  in  elastic  solids  may  be  properly  called  viscosity  of 
solids,  because,  as  being  an  internal  resistance  to  change  of  shape 
depending  on  the  rapidity  of  the  change,  it  must  be  classed  with 
fluid  molecular  friction,  which  by  general  consent  is  called  viscosity  of 
fluids.  But,  at  the  same  time,  we  feel  bound  to  remark  that  the  word 
viscosity,  as  used  hitherto  by  the  best  writers,  when  solids  or  hetero- 
geneous semisolid-semifluid  masses  are  referred  to,  has  not  been 
distinctly  applied  to  molecular  friction,  especially  not  to  the  molecular 
friction  of  a  highly  elastic  solid  within  its  limits  of  high  elasticity,  but 
has  rather  been  employed  to  designate  a  property  of  slow,  continual 
yielding  through  very  great,  or  altogether  unlimited,  extent  of  change 
of  shape,  under  the  action  of  continued  stress.  It  is  in  this  sense 
that  Forbes,. for  instance,  has  used  the  word  in  stating  that  'Viscous 
Theory  of  Glacial  Motion'  which  he  demonstrated  by  his  grand 
observations  on  glaciers.  As,  however,  he,  and  many  other  writers 
after  him,  have  used  the  words  plasticity  and  plastic,  both  with  refer- 
ence to  homogeneous  solids  (such  as  wax  or  pitch,  even  though  also 
brittle;  soft  metals;  etc.),  and  to  heterogeneous  semisolid-semifluid 
masses  (as  mud,  moist  earth,  mortar,  glacial  ice,  etc.),  to  designate 
the  property^  common  to  all  those  cases,  of  experiencing,  under 
continued  stress  either  quite  continued  and  unlimited  change  of  shape, 
or  gradually  very  great  change  at  a  diminishing  (asymptotic)  rate 
through  infinite  time  ;  and  as  the  use  of  the  term  plasticity  implies  no 
more  than  does  viscosity,  any  physical  theory  or  explanation  of  the' 
property,  the  word  viscosity  is  without  inconvenience  left  available 
for  the  definition  we  have  given  of  it  above. 

684.  K  perfect  fluid,  or  (as  we  shall  call  it)  a  fluid,  is  an  unrealizable 
conception,  like  a  rigid,  or  a  smooth,  body:  it  is  defined  as  a  body 
incapable  of  resisting  a  change  of  shape  :  and  therefore  incapable  of 
experiencing  distorting  or  tangential  stress  (§  640).  Hence  its  pres- 
sure on  any  surface,  whether  of  a  solid  or  of  a  contiguous  portion  o^ 

^  See  Froceci'/uigs  of  the  Royal  Socuiy,  May  1865,  'On  the  Viscosity  and 
Elasticity  of  Metals'  (W.  Thomson). 

'■  Some  confusion  of  ideas  might  have  been  avoided  on  the  part  of  writers  who 
have  professedly  objected  to  Forbes'  theory  while  really  objecting  only  (and  we 
■believe  groundlessly)  to  his  usage  of  the  word  viscosity,  if  they  had  paused  to 
consider  that  no  one  physical  explanation  can  hold  for  those  several  cases;  and 
that  Forbes'  theory  is  merely  the  proof  by  observation  that  glaciers  have  the 
property  that  niud  (heterogeneous),  mortar  (heterogeneous),  pitch  (homogeneous), 
water  (homogeneous),  all  have  of  changing  shape  indefinitely  and  continuously 
under  the  action  of  continued  stress. 


STATICS   OF  SOLIDS  AND   FLUIDS.  265 

the  fluid,  fs  at  every  point  perpendicular  to  the  surf"ice.  In  equi- 
librium, all  common  liquids  and  gaseous  fluids  fulfil  the  definition. 
But  there  is  finite  resistance,  of  the  nature  of  friction,  opp.osing  change 
of  shape  at  a  finite  rate;  and,  therefore,  while  a  fluid  is  changing 
shape,  it  exerts  tangential  force  on  every  surface  other  than  normal 
planes  of  the  stress  (§  635)  required  to  keep  this  change  of  shape 
going  on.  Hence;  although  the  hydrostatical  results,  to  which  we 
immediately  proceed,  are  verified  in  practice;  in  treating  of  hydro- 
kinetics,  in  a  subsequent  chapter,  we  shall  be  obliged  to  introduce  the 
consideration  of  fluid  friction,  except  in  cases  whurc  the  circumstances 
are  such  as  to  render  its  eft'ects  insensible. 

685.  With  reference  to  a  fluid  the  pressure  at  any  pcbd  in  any 
direction  is  an  exjjression  used  to  denote  the  average  pressure  per  unit 
of  area  on  a  plane  surface  imagined  as  containing  the  point,  and 
perpendicular  to  the  direction  in  question,  when  the  area  of  that 
surface  is  indefinitely  diminished. 

686.  At  any  point  in  a  fluid  at  rest  the  pressure  is  the  same  in 
all  directions:  and,  if  no  external  forces  act,  the  pressure  is  the  same 
at  every  point.  For  the  proof  of  these  and  most  of  the  following 
propositions,  we  imagine,  according  to  §  584,  a  definite  portion  of 
the  fluid  to  become  solid,  without  changing  its  mass,  form,  or 
dimensions. 

Suppose  the  fluid  to  be  contained  in  a  closed  vessel,  the  pressure 
within  depending  on  the  pressure  exerted  on  it  by  the  vessel,  and  not 
on  any  external  force  such  as  gravity. 

687.  The  resultant  of  the  fluid  pressures  on  the  elements  of  any 
portion  of  a  spherical  surface  must,  like  each  of  its  components,  pass 
through 'the  centre  of  the  sphere.  Hence,  if  we  suppose  {§  584)  a 
portion  of  the  fluid  in  the  form  of  a  plano-convex  lens  to  be  solidified, 
the  resultant  pressure  on  the  plane  side  must  pass  through  the  centre 
of  the  sphere;  and,  therefore,  being  perpendicular  to  the  plane,  must 
pass  through  the  centre  of  the  circular  area.  From  this  it  is  obvious 
that  the  pressure  is  the  same  at  all  points  of  any  plane  in  the  fluid. 
Hence  the  resultant  pressure  on  any  plane  surface  passes  through 
its  centre  of  inertia. 

Next,  imagine  a  triangular  j)rism  of  the  fluid,  witli  ends  perpen- 
dicular to  its  faces,  to  be  solidified.  The  resultant  j^ressurcs  on  its 
ends  act  in  the  line  joining  the  centres  of  inertia  of  their  areas, 
and  are  equal  since  the  resultant  pressures  ovi  th.e  >le'.es  are  m 
directions  perpendicular  to  this  line.  Hence  the  pressure  is  'Ci\c  same 
in  all  parallel  planes. 

But  the  centres  of  inertia  of  the  three  faces,  and  the  resultant 
pressures  applied  there,  lie  in  a  triangular  section  parallel  to  the  ends. 
The  pressures  act  at  the  middle  points  of  the  sides  of  this  triangle, 
and  perpendicularly  to  them,  so  that  theix  directions  meet  in  a 
point.  And,  as  they  are  in  equilibrium,  they  must  be  proijortio'ial 
to  the  respective  sides  of  the  triangle;  that  is,  to  tl-.e  bread'h  ;,  or 
areas,  of  the  laces  of  the  pritm.     Thus  the  resultant  pressures  on  the 


266  ABSTRACT  DYNAMICS. 

faces  must  be  proportional  to  the  areas  of  the  faces,  and  therefore 
the  pressure  is  equal  in  any  two  planes  which  meet. 

Collecting  our  results,  we  see  that  the  pressure  is  the  same  at  all 
points,  and  in  all  directions,  throughout  the  fluid  mass. 

688.  Hence  if  a  force  be  applied  at  the  centre  of  inertia  of  each 
face  of  a  polyhedron,  with  magnitude  proportional  to  the  area  of 
the  face,  the  polyhedron  will  be  in  equilibrium.  For  we  may  suppose 
the  polyhedron  to  be  a. solidified  portion  of  the  fluid.  The  resultant 
pressure  on  each  face  will  then  be  proportional  to  its  area,  and  will 
act  at  its  centre  of  inertia;  which,  in  this  case,  is  the  Centre  of 
Pressure. 

689.  Another  proof  of  the  equality  of  pressure  throughout  a  mass 
of  fluid,  uninfluenced  by  other  external  force  than  the  pressure  of  the 
containing  vessel,  is  easily  furnished  by  the  energy  ciiterion  of  equi- 
librium, §  254;  but,  to  avoid  complication,  we  will  consider  the  fluid 
to  be  incompressible.  Suppose  a  number  of  pistons  fitted  into 
cylinders  inserted  in  the  sides  of  the  closed  vessel  containing  the  fluid. 
Then,  if  A  be  the  area  of  one  of  these  pistons,  p  the  average  pressure 
on  it,  ^  the  distance  through  which  it  is  pressed,  in  or  out;  the  energy 
criterion  is  that  no  work  shall  be  done  on  the  whole,  i.  e.  that 

Aj^x^  +  Aj^x^  +  ..  ^2  {A^x)  =  o, 
as  much  work  being  restored  by  the  pistons  which  are  forced  out,  as 
is  done  by  those  forced  in.     Also,  since  the  fluid  is  incompressible,  it 
must  have  gained  as  much  space  by  forcing  out  some  of  the  pistons 
as  it  lost  by  the  intrusion  of  the  others.     This  gives 

A^x^-¥A^x^  +  ...='2,{Ax)^o. 
The  last  is  the  only  condition  to  which  x^,  x^,  etc.,  in  the  first  equa- 
tion, are  subject;  and  therefore  the  first  can  only  be  satisfied  if 

A=A=A  =  etc., 
that  is,  if  the  pressure  be  the  same  on  each  piston.     Upon  this  pro- 
perty depends  the  action  of  Bramah's  Hydrostatic  Press. 

If  the  fluid  be  compressible,  the  work  expended  in  compressing  it 
from  volume  Vio  V-SV,  at  mean  pressure/,  is/SK 

If  in  this  case  we  assume  the  pressure  to  be  the  -same  throughout, 
we  obtain  a  result  consistent  with  the  energy  criterion. 

The  work  done  on  the  fluid  is  2  {Apx),  that  is,  in  consequence  of 
the  assumption,  /2  {Ax). 

But  this  is  equal  to  pZ  V, 

for,  evidently,  2  {Ax)  -  8  V. 

690.  When  forces,  such  as  gravity,  act  from  external  matter  upon 
the  substance  of  the  fluid,  either  in  proportion  to  the  density  of  its 
own  substance  in  its  different  parts,  or  in  proportion  to  the  density 
of  electricity,  or  of  magnetic  polarity,  or  of  any  other  conceivable 
accidental  property  of  it,  the  pressure  will  still  be  the  same  in  all 
directions  at  any  one  point,  but  will  now  vary  continuously  from 
point  to  point     For  the  preceding  demonstration  (§  687)  may  still 


STATICS   OF  SOLIDS  AXD   FLUIDS.  267 

be  applied  by  simply  taking  the  dimensions  o^  the  prism  small 
enough;  since  the  pressures  are  as  the  squares  of  its  linear  dimen- 
sions, and  the  effects  of  the  applied  forces  such  as  gravity,  as  the 
cubes. 

691.  When  forces  act  on  the  whole  fluid,  surfaces  of  equal  pressure, 
if  they  exist,  must  be  at  every  point  perpendicular  to  the  direction  of 
the  resultant  force.  For,  any  prism  of  the  fluid  so  situated  that  the 
whole  pressures  on  its  ends  are  equal  must  experience  from  the 
applied  forces  no  component  in  the  direction  of  its  length;  and, 
therefore,  i£  the  prism  be  so  small  that  from  point  to  point  of  it  the 
direction  of  the  resultant  of  the  applied  forces  does  not  vary  sensibly, 
this  direction  must  be  perpendicular  to  the  length  of  the  prism. 
From  this  it  follows  that  whatever  be  the  physical  origin,  and  the  lav/, 
of  the  system  of  forces  acting  on  the  fluid,  and  whether  it  be  con- 
servative or  non-conservative,  the  fluid  cannot  be  in  equilibrium  unless 
the  lines  of  force  possess  the  geometrical  property  of  being  at  right 
angles  to  a  series  of  surfaces. 

C9_2.  Again,  considering  two  surfaces  of  equal  pressure  infinitely 
near  one  another,  let  the  fluid  between  them  be  divided  into  columns 
of  equal  transverse  section,  and  having  their  lengths  perpendicular  to 
the  surfaces.  The  difference  of  pressure  on  the  two  ends  being  the 
same  for  each  column,  the  resultant  applied  forces  on  the  fluid  masses 
composing  tliem  must  be  equal.  Comparing  this  with  §  506,  v,-e  see 
that  if  the  applied  forces  constitute  a  conservative  system,  the  density 
of  matter,  or  electricity,  or  whatever  property  of  the  substance  they 
depend  on,  must  be  ec^ual  throughout  the  layer  under  consideration. 
This  is  the  celebrated  hydrostatic  proposition  that  m  a  fluid  at  rcsf, 
surfaces  of  equal  pressure  are  also  surfaces  of  equal  de/isi/y  and  of  equal 
potential. 

6S3.  Hence  when  gravity  is  the  only  external  force  considered, 
surfaces  of  equal  pressure  and  equal  density  are  (when  of  moderate 
extent)  horizontal  planes.  On  this  depends  the  action  of  levels, 
siphons,  barometers,  etc. ;  also  the  separation  of  liquids  of  different 
densities  (which  do  not  mix  or  combine  chemically)  into  horizontal 
strata,  etc.,  etc.  The  free  surface  of  a  liquid  is  exposed  to  the  pressure 
of  the  atmosphere  simply;  and  therefore,  when  in  equilibrium,  must 
be  a  surface  of  equal  pressure,  and  consequently  level.  In  extensive 
sheets  of  water,  such  as  the  American  lakes,  differences  of  atmo- 
spheric pressure,  even  in  moderately  calm  weather,  often  produce  con- 
siderable deviations  from  a  truly  level  surface. 

C94.  The  rate  of  increase  of  pressure  per  unit  of  leneth  in  the 
direction  of  the  resultant  force,  is  equal  to  the  intensity  of  ilie  force 
reckoned  per  unit  of  volume  of  the  tluid.  Let  F  be  the  resultant 
force  per  unit  of  volume  in  one  of  the  columns  of  i^  692;  /  and/' 
the  pressures  at  the  ends  of  the  column,  /  its  length,  6"  its  section.-  We 
have,  for  the  equilibrium  of  the  column, 

{p'-p)S^SlF 
Hence  the  rate  of  increase  of  pressure  per  uniLof  length  is  F. 


263  ABSTRACT  DYNAMICS. 

If  the  applied  forces  belong  to  a  conservative  system,  for  which 
V  and  V  are  the  values  of  the  potential  at  the  ends  of  the  column, 
we  have  (§504) 

V'-V=---  IFp, 
where  p  is  the  density  of  the  fluid.     This  gives 

or  dp  =  —pdV. 

Hence  in  the  case  of  gravity  as  the  only  impressed  force  the  rate 
of  increase  of  pressure  per  unit  of  depth  in  the  fluid  is  p,  in  gravitation 
measure  (usually  employed  in  hydrostatics).  In  kinetic  or  absolute 
measure  (§  189)  it  is  ^p. 

If  the  fluid  be  a  gas,  such  as  air,  and  be  kept  at  a  constant  tem- 
perature, we  have  p  =  cp,  where  c  denotes  a  constant,  the  reciprocal  of 
H,  the  'height  of  the  homogeneous  atmosphere,'  defined  (§  695) 
below.  Hence,  in  a  calm  atmosphere  of  uniform  temperature  we 
have 

^^  =  -cdV; 

P 
and  from  this,  by  integration, 

where  p^  is    the  pressure  at  any  particular  level  (the  sea-level,  for 
instance)  where  we  choose  to  reckon  the  potential  as  zero. 

When  the  differences  of  level  considered  are  infinitely  small  in 
comparison  with  the  earth's  radius,  as  we  may  practically  regard  them, 
in  measuring  the  heights  of  mountains,  or  of  a  balloon,  by  the  baro- 
meter, the  force  of  gravity  is  constant,  and  therefore  differences  of 
potential  (force  being  reckoned  in  units  of  weight)  are  simply  equal 
to  differences  of  level.  Hence  if  x  denote  height  of  the  level  of 
pressure/  above  that  of Z^,,  we  have,  in  the  preceding  formulae,  V~x, 
and  therefore 

p  =  p„(~";  that  is, 

695.  If  the  air  be  at  a  constant  temperature,  the  pressure 
diminishes  in  geometrical  progression  as  the  height  increases  in 
arithmetical  progression.  This  theorem  is  due  to  Halley.  Without 
formal  mathematics  we  see  the  truth  of  it  by  remarking  that  dif- 
ferences of  pressure  are  (§  694)  equal  to  differences  of  level  multiplied 
by  the  density  of  the  fluid,  or  by  the  proper  mean  density  when  the 
density  differs  sensibly  between  the  two  stations.  But  the  density, 
when  the  temperature  is  constant,  varies  in  simple  proportion  to 
the  pressure,  according  to  Eoyle's  law.  Hence  differences  of  pres- 
sure between  pairs  of  stations  differing  equally  in  level  are  pro- 
portional to  the  proper  mean  values  of  the  whole  pressure,  which  is 
the  well-known  compound  interest  law.  The  rate  of  diminution  of 
pressure  per  unit  of  length  upwards  in  proportion  to  the  whole 
pressure  at  any  point,  is  of  course  equal  to  the  reciprocal  of  the  height 
above  that  point  that  the  atmosphere  must  have,  if  of  constant 
density,  to  give  that  pressure  by  its  weight.  The  height  thus  defined 
is <;ommonly  called  'the  height  of  the  homogeneous  atmosphere,'  a 


STATICS  OF  SOLIDS  AND  FLUIDS  269 

very  convenient  conventional  expression.  It  is  equal  to  the  product 
of  the  volume  occupied  by  the  unit  mass  of  the  gas  at  any  pressure 
into  the  value  of  that  pressure  reckoned  per  unit  of  area,  in  terms  of 
the  weight  of  the  unit  of  mass.  If  we  denote  il  by  ^,  the  expo- 
nential expression  of  the  law  is 

which  agrees  with  the  final  formula  of  §  694. 

The  value  of  H  for  dry  atmospheric  air,  at  the  freezing  tem- 
perature, according  to  Regnault,  is,  in  the  latitude  of  Paris,  799,020 
centimetres,  or  26,215  f<^*^t.  Being  inversely  as  the  force  of  gravity 
in  different  latitudes  (§  187),  it  is  798,533  centimetres,  or  26,199  '^'^^''■y 
in  the  latitude  of  Edinburgh  and  Glasgow. 

696.  It  is  both  necessary  and  sufficient  for  the  equilibrium  or  an 
incompressible  fluid  completely  filling'  a  rigid  closed  vessel,  and 
influenced  only  by  a  conservative  system  of  forces,  that  its  density  be 
uniform  over  every  equipotential  surface,  that  is  to  say,  every  surface 
cutting  the  lines  of  force  at  right  angles.  If,  however,  the  boundary, 
or  any  paVt  of  the  boundary,  of  the  fluid  mass  considered,  be  not 
rigid;  whether  it  be  of  flexible  solid  matter  (as  a  membrane,  or  a  thin 
sheet  of  elastic  solid),  or  whether  it  be  a  mere  geometrical  boundary, 
on  the  other  side  of  which  there  is  another  fluid,  or  ?wthing  [a  case 
which,  without  believing  in  vacuum  as  a  reality,  we  may  admit  in 
abstract  dynamics  (§  391)],  a  farther  condition  is  necessary  to  secure 
that  the  pressure  from  without  shall  fulfil  the  hydrostatic  equation 
at  every  ])oint  of  the  boundary.  In  the  case  of  a  bounding  membrane, 
this  condition  must  be  fulfilled  either  through  pressure  artificially 
applied  from  without,  or  through  the  interior  elastic  forces  of  the 
matter  of  the  membrane.  In  the  case  of  another  fluid  of  different 
density  touching  it  on  the  other  side  of  the  boundary,  all  round  or 
over  some  part  of  it,  with  no  separating  membrane,  the  condition 
of  equilibrium  of  a  heterogeneous  fluid  is  to  be  fulfilled  relatively 
to  the  whole  fluid  mass  made  up  of  the  two;  which  shows  that  at  the 
boundary  the  pressure  must  be  constant  and  equal  to  that  of  the  fluid 
on  the  other  side.  Thus  water,  oil,  mercury,  or  any  other  liquid,  in 
an  open  vessel,  with  its  free  surface  exposed  to  the  air,  requires  for 
equilibrium  simjily  that  this  surface  be  level. 

697.  Recurring  to  the  consideration  of  a  finite  mass  of  fluid 
completely  filling  a  rigid  closed  vessel,  we  sec,  froni  what  prece^lcs, 
that,  if  homogeneous  and  incompressible,  it  cannot  be  disturbed  from 
equilibrium  by  any  conservative  systen^i  of  forces;  but  wc  do  not 
require  the  analytical  investigation  to  prove  this,  as  we  should  have 
'the  perpetual  motion'  if  it  were  denied,  wliich  would  violate  the 
hypothesis  that  the  system  of  forces  is  conservative.  On  the  other 
hand,  a  non-conservative  system  of  forces  cannot,  under  any  circum- 
stances, equilibrate  a  fluid  which  is  either  uniform  in  density  through- 
cut,  or  of  homogeneous  substance,  rendered  heterogeneous  in  density 
.only  through  difference  of  pressure.     But  if  the  forces,  though  not 


270  ABSTRACT  DYNAMICS. 

conservative,  be  such  that  through  every  point  of  the  space  occupied 
by  the  fluid  a  surface  may  be  dxawn  which  shall  cut  at  right  angjes 
all  the  lines  of  force  it  meets,  a  heterogeneous  fluid  will  rest  in 
equilibrium  under  their  influence,  provided  (§  692)  its  density,  from 
ppint  to  point  of  every  one  of  these  orthogonal  surfaces,  varies  in- 
veugely  as  the  product  of  the  resultant' fprce  into  the  thickness  of 
the  infinitely  thin  layer  of  space  between  that  surface  and  another  of 
the  orthogonal  surfaces  infiniteljj.  near  it  on  either  side.  (Compare 
§  S06). 

698.  If  we  imagine  all  the  fluid  to  become  rigid  except  an  infinitely 
thin  clo5ed  tubular  portion  lying  in  a  surface  of  equal,  density,  and  if 
the  fluid  in  this  tubular  circuit  be  moved  any  length  along-  the  tube 
and  left  at  rest,  it  will  remain  in  equilibrium  in  the  new  position, 
all  positions  of  it  in  the  tube  being  indifferent  because  of  its  homo- 
geneou3n4SS.  Hence  the  work  (positive  or  negative)  done  by  the 
force  {Xy  y,  Z)  on  any  portion  of  the  fluid  in  any  displacement 
along  the  tube  is  balanced  by  the  wofk  (negative  or  positive)  done  on 
the  remainder  of  the  fluid  in  the  tube.  Hence  a  single  particle,  acted 
on  only  by  X,  V,  Z,  while  moving  round  the  circuit,  that  is  moving 
along  any  closed  curve  on  a  surface  of  equal  density,  has,  at  the  end 
of  one  complete  circuit,  done  just  as  much  work  against  the  force  in 
some  parts  of  its  course,  as  the  forces  have  done  on  it  in  the  re- 
mainder of  the  circuit. 

699.  The  following  imaginary  example,  and  its  realization  in  a 
subsequent  section  (§  701),  show  a  curiously  interesting  practical 
application  of  the  theory  of  fluid  equilibrium  under  extraordinary 
circumstances,  generally  regarded  as  a  merely,  abstract  analytical 
theory,  practically  useless  and  quite  unnatural,  'because  forces  in 
nature  follow  the  conservative  law.' 

700!  Let  the  lines  of  force  be  circles,  with  their  centres  all  in  one 
line,  and  their  planes  perpendicular  to  it.  They  are  cut  at /ight 
angles  by  planes  through  this  axis;  and  therefore  a  fluid  may  be  in 
equilibrium  under  such  a  system  of  forces.  The  system  will  not  be 
conservative  if  the  intensity  of  the  force  be  according  to  any  other  law 
than  inverse  proportionality  to  distance  from  this  axial  line;  and  the 
fluid,  to  be  in  equilibrium,  must  be  heterogeneous,  and  be  so  dis- 
tributed as  to  vary  in  density  from  point  to  point  of  every  plane 
through  the  axis,  inversely  as  the  product  of  the  force  into  the 
distance  from  the  axis.  But  from  one  such  plane  to  another  it  may 
be  cither  uniform  in  density,  or  may  vary  arbitrarily.  To  particularize 
farther,  we  may  suppose  the  force  to  be  in  direct  simple  proportion 
to  the  distance  from  the  axis.  Then  the  fluid  will  be  in  equilibrium 
if  its  density  varies  from  point  to  point  of  every  plane  through  the 
axis,  inversely  as  the  square  of  that  distance.  If  we  still  farther 
particularize  by  making  the  force  uniform  all  round  each  circular  line 
of  force,  the  distribution  bf  force  becomes  precisely  that  of  the  kinedc 
reactions  of  the  parts  of  a  rigid  body  against  accelerated  rotation. 
The  fluid  pressure  will 


STATICS  OF  SOLIDS  AND  FLUIDS.  271 

axis.  And  in  one  such  plane,  which  we  may  imagine  carried  round 
the  axis  in  the  direction  of  the  force,  the  fluid  pressure  will  increase  in 
sinriple  proportion  to  the  angle  at  a  rate  per  un;t  angle  (§  55)  equal  to 
the  product  of  the  density  at  unit  distance  into  the  force  at  unit  distance. 
Hence  it  must  be  remarked,  that  if  any  closed  line  (or  circuit)  can  be 
drawn  round  the  axis,  widiout  leaving  the  fluid,  there  cannot  be 
equilibrium  without  a  firm  partition  cutting  every  such  circuit,  and 
maintaining  the  difference  of  pressures  on  the  two  sides  of  it,  corre- 

spoading  to   the  angle  jtt.     Thus,  if  the  

axis   pass  through   the  fluid  in  any  part,  /i^    "^X 

there  must  be  a  partition  extending  from  V^J^**^'**^   \ 

this  part  of  the  axis  continuously  to  the       ^^f!^^*'^^jr  m   ^ 

outer  bounding  surface  of  the  fluid.     Or  if    ^^    :-^       .  ^  -<^§ 
the  bounding  surface  of  the  whole  fluid  be     ^       '^^W^^'*^^-'^''^ 
annular  (like  a   hollow  anchor-ring,  or  of     ^^^^^        *v\      -^^ 
any  irregular  shape),  in  other  words,  if  the        ^^^jz--     ''^^i^^ 
fluid  fills  a   tubular  circuit;    and  the  axis  ^^^i£^M^j^ 

(A)  pass  through  the  aperture  of  the  ring 

(without  passing  into  the  fluid);  there  must  be  a  firm  partition  (CD) 
extending  somewhere  continuously  across  the  channel,  or  possagS'  or 
tube,  to  stop  the  circulation  of  the  fluid  round  it;  otherwise  there 
could  not  be  equilibrium  with  the  supposed  forces  in  action.  If  we 
further  suppose  the  density  of  the  fluid  to  be  uniform  round  each  of 
the  circular  hnes  offeree  in  the  system  we  have  so  far  considered  (so 
that  the  density  shall  be  equal  over  every  circular  cylinder  having  the 
line  of  their  centres  for  its  axis,  and  shall  vary  from  one  such 
cylindrical  surface  to  another,  inversely  as  the  squares  of  their  radii), 
we  may,- without  disturbing  the  equilibrium,  impose  any  conservative 
system  of  force  in  lines  perpendicular  to  the  axis;  that  is  (§  506),  any 
system  of  force  in  this  direction,  with  intensity  varying  as  some 
function  of  the  distance.  If  this  function  be  the  simple  distance,  the 
superimposed  system  of  force  agrees  precisely  with  the  reactions 
against  curvature,  that  is  to  say,  the  centrifugal  forces,  of  the  parts  of 
a  rotating  rigid  body. 

701.  Thus  we  arrive  art  tlic  remarkable  conclusion,  that  if  a  ngid 
closed  box  be  coi"n;jL:Lly  filk-ci  wi'.h  incompressible  hctL-rogeneous 
fluid,  of  dcr.sity  varving  inversely  as  the  square  of  the  distance  from 
a  certain  line,  and  if  the  box  be  movable  round  this  line  as  a  fixed 
axis,'  arid  ije  urged  in  any  way  by  furces  applied  to  its  outside,  the 
fluid  will  remain  in  ciiuihljriuin  relati\e!y  to  the  box;  that  is  to  say, 
will  move  round  v.-itii  th.e  box  as  if  the  whole  were  one  rigid  body, 
and  win  come  to  re.-i  v.itli  th.e  box  if  the  box  be  brougiu  again  to 
rest:  provided  always  the  jTereding  con-liiiLin  as  to  partitions  be 
fulfilled  if  the  axis  pass  through  tlie  lluid,  or  be  surrounded  ly 
continuous  lines  of  fluid.  For,  in  starting  from  re^t,  {/  the  fluid 
moves  like  a  rigid  solid,  wc  have  leaotions  against  acceleration, 
tangential  to  the  circles  of  motion,  ami  equal  m  amount  fo  ui'-  per 
unit  of  mass  of  the  fluid  at  distance  r  tVorn  the  axis,  w  I'einj;  tlie  rate 


272  ABSTRACT  DYNAMICS, 

of  acceleration  (§  57)  of  the  angular  velocity ;  and  (see  Vol.  II.)  we 
have,  in  the  direction  perpendicular  to  the  axis  outwards,  reaction 
against  curvature  of  path,  that  is  to  say,  'centrifugal  force,'  equal  to 
<i>V  per  unit  of  mass  of  the  fluid.  Hence  the  equilibrium  which  we 
have  demonstrated  in  the  preceding  section,  for  the  fluid  supposed 
at  rest,  and  arbitrarily  influenced  by  two  systems  of  force  (the  circular 
non-conservative  aftd  the  radial  conservative  system)  agreeing  in  law 
with  these  forces,  of  kinetic  reaction,  proves  for  us  now  the  D'Alem- 
bert  (§  23o)'equilibrium  condition  for  the  motion  of  the  whole  fluid  as 
of  a  rigid  body  experiencing  accelerated  rotation:  that  is  to  say, 
shows  that  this  kind  of  motion  fulfils  for  the  actual  circumstances  the 
laws  of  motion,  and,  therefore,  that  it  is  the  motion  actually  taken  by 
the  fluid. 

702.  In  §  688  we  considered  the  'resultant  pressure  on  a  plane 
surface,  when  the  pressure  is  uniform.  We  may  now  consider 
(briefly)  the  resultant  pressure  on  a  plane  area  when  the  pressure 
varies  from  point  to  point,  confining  our  attention  to  a  case  of 
great  importance; — that  in  which  gravity  is  the  only  applied  force, 
and  the  fluid  is  a  nearly  incompressible  Hquid  such  as  water.  In  this 
case  the  determination  of  the  position  of  the  Centre  of  Pressure  is 
very  simple;  and  the  whole  pressure  is  the  same  as  if  the  plane  area 
were  turned  about  its  centre  of  inertia  into  a  horizonal  position. 

The  pressure  at  any  point  at  a  depth  z  in  the  liquid  may  be  ex- 
pressed by 

where  p  is  the  (constant)  density  of  the  liquid,  and  p^  the  (atmo- 
spheric) pressure  at  the  free  surface,  reckoned  in  units  of  weight  per 
unit  of  area. 

Let  the  axis  of  x  be  taken  as  the  intersection  of  the  plane  of  the 
immersed  plate  with  the  free  surface  of  the  liquid,  and  that  of  y 
perpendicular  to  it  and  in  the  plane  of  the  plate.  Let  a  be  the 
inclinaticJh  of  the  plate  to  the  vertical.  Let  also  A  be  the  area  of  the 
portion  of  the  plate  considered,  and  x,y,  the  co-ordinates  of  its  centre 
of  inertia. 

Then  the  whole  pressure  is 

jjpdxdy  =  //  (/„  +  py  cos  o)  dxdy 
-  Ap^  -f  Apycos  a. 
The  moment  of  the  pressure  about  the  axis  of  x  is 
/  [pydxdy  =  Ap^y  +  Ak'p  cos  a, 

/''  being  the  radius  of  gyration  of  the  plane  area  about  the  axis  oi  x. 
For  the  moment  about  _>'  we  have 

j ^pxdxdy  =  yipjx  +  p  cos  a  JJxydxdy. 

The  first  terms  of  these  three  expressions  merely  give  us  again  the 
results  of  §  688;  we  may  therefore  omit  them.  This  will  be  equi- 
valent to  introducing  a  stratum  of  additional  liquid  above  the  free 
surface  such  as  to  produce  an  equivalent  to  the  atmospheric  pressure. 


STATICS  OF  SOLIDS  AND  FLUIDS.  273 

If  the  origin  be  now  shifted  to  the  upper  surface  of  this  stratum  we 
have 

Pressure  =  Apy  cos  a, 

Moment  about  Ox  =  Ak'p  cos  a, 
Distance  of  centre  of  oressure  from  axis  of  -r  =  — . 

y 

But  if  k  be  the  radius  of  g}'ration  of  the  plane  area  about  a  horizontal 
axis  in  its  plane,  and  passing  through  its  centre  of  inertia,  we  have 

Hence  the  distance,  measured  parallel  to  the  axis  of  ^,  of  the  centre 
of  pressure  from  the  centre  of  inertia  is 

y' 

and,  as  we  might  expect,  diminishes  as  the  plane  area  is  more  and 

more  submerged.  If  the  plane  area  be  turned  about  the  line  through 
its  centre  of  inertia  parallel  to  the  axis  of  .v,  this  distance  varies  as 
the  cosine  of  its  inclination  to  the  vertical;  supposing,  of  course,  that 
by  the  rotation  neither  more  nor  less  of  the  plane  area  is  submerged. 

703.  A  body,  wholly  or  partially  immersed  in  any  fluid  influenced 
by  gravity,  loses,  through  fluid  pressure,  in  apparent  weight  an  amount 
equal  to  the  weight  of  the  fluid  displaced.  For  if  the  body  were 
removed,  and  its  place  filled  with  fluid  homogeneous  with  the  sur- 
rounding fluid,  there  would  be  equilibrium,  even  if  this  fluid  be  sup- 
posed to  become  rigid.  And  the  resultant  of  the  fluid  pressure  upon 
it  is  therefore  a  single  force  equal  to  its  weight,  and  in  the  vertical 
line  through  its  certre  of  gravity.  Bat  the  fluid  pressure  on  the 
originally  immersed  body  was  the  same  all  over  as  on  the  solidified 
portion  of  fluid  by  which  for  a  moment  we  have  imagined  it  replaced, 
and  therefore  must  have  the  same  resultant.  This  proposition  is  of 
great  use  in  Hydrometry,  the  determination  of  specific  gravity,  etc, 
etc. 

704.  The  following  lemma,  while  in  it?elf  interesting,  is  of  great 
use  in  enabling  us  to  simplify  the  succeeding  investigations  regarding 
the  stability  of  equilibrium  of  floating  bodies: — 

Let  a  homogeneous  solid,  the  weight  of  unit  of  volume  of  which 
we  suppose  to  be  unity,  be  cut  by  a  horizontal  plane  in  XYX  Y. 
Let  O  be  the  centre  of  inertia, 
and  let  XX',  VV  be  the  principal 
axes,  of  this  area. 

Let  there  be  a  second  plane 
section  of  the  solid,  through  VY, 
inclined  to'  the  first  at  an  infinitely  ^Y' 
small  angle,  0.  Then  (i)  the 
volumes  of  the  two  wedges  cut 
from  the  solid  by  these  sections 
are  equal;  (2)  their  centres  of 
inertia   he   in  one  plane  perpen- 


374  ABSTRACT  DYNAMICS. 

dicular  XoYY',  and  (3)  the  moment  of  the  weight  of  each  of  these, 
round  YY,  is  equal  to  the  moment  of  inertia  about  it  of  the  corre- 
sponding portion  of  the  area  multiplied  by  B. 

Take  OX,  OY  z.%  axes,  and  let  6  be  the  angle  of  the  wedge :  the 
thickness  of  the  wedge  at  any  point  P,  {x,  y),  is  6x,  and  the  volume 
of  a  right  prismatic  portion  whose  base  is  the  elementary  area  dxdy 
at  P  is  Oxdxdy. 

Now  let  [  ]  and  ( )  be  employed  to  distinguish  integrations  extended 
over  the  portions  of  area  to  the  right  and  left  of  the  axis  oi  y  re- 
spectively, while  integrals  over  the  whole  area  have  no  such  distin- 
guishing mark.  Let  v  and  v'  be  the  volumes  of  the  wedges ;  (x,  J'), 
(^'jj-')  the  co-ordinates  of  their  centres  of  inertia.  Then 
v=^0  l\\xdxdy^ 
~v'  =  0  Ijjxdxdy), 

whence  v-d  =  &  Jjxdxdy  =  o  since  O  is  the  centre  of  inertia.    Hence 
v  —  v\  which  is  (i). 
Again,  taking  moments  about  XX\ 

7Jy  =  6\Jixydxdy], 
and  -  z''y'  =  6  (Jjxydxdy). 

Hence  z/y  -  v'y'  —  $  jjxydxdy. 

But  for  a  principal  axis  ^xydm  vanishes.  Hence  ziy-vy  =  o, 
whence,  since  v  ~  v',  we  have 

y—y%  which  proves  (2). 

And  (3)  is  merely  a  statement  in  words  of  the  obvious  equation 

UJx.xOdxdy]  -  0  [JJx^.dxdy]. 

705.  If  a  positive  amount  of  work  is  required  to  produce  any 
possible  infinitely  small  displacement  of  a  body  from  a  position  of 
equilibrium,  the  equilibrium  in  this  position  is  stable  (§  256).  To 
apply  this  test  to  the  case  of  a  floating  body,  we  may  remark,  first, 
tliat  any  possible  infinitely  small  displacement  may  (§§  30,  106)  be 
conveniently  regarded  as  compounded  of  two  horizontal  displacements 
in  lines  at  right  angles  to  one  another,  o;"i.e  vertical  displacement,  and 
three  rotations  round  rectangular  axes  through  any  chosen  point.  If 
one  of  these  axes  be  vertical,  then  three  of  the  component  displace- 
ments, viz.  the  two  horizontal  displacements  and  the  rotation  about 
the  vertical  axis,  require  no  work  (positive  or  negative),  and  therefore, 
so  far  as  they  are  concerned,  the  equilibrium  is  essentially  neutral. 
But  so  far  as  the  other  three  modes  of  displacement  are  concerned, 
the  equilibrium  may  be  positively  stable,  or  may  be  unstable,  or  may 
be  neutral,  according  to  the  fulfilment  of  conditions  which  we  now 
proceed  to  investigate. 

706.  If,  first,  a  simple  vertical  displacement,  downwarfls,  let  us 
suppose,  be  made,  the  work  is  done  against  an  increasing  resultant 
of  upward  fluid  pressure,  and  is  of  course  equal  to  the  mean  increase 
of  this  force  multiplied  by  the  whole  space.  If  this  space  be  denoted 
by  z,  the  area  of  the  plane  of  flotation  by  A,  and  the  weight  of  unit 
bulk  of  the  liquid  by  w,  the  increased  bulk  of  immersion  is  clearly  y4». 


STATICS   OF  SOLIDS  AND  FLUIDS. 


275 


and  therefore  the  increase  of  the  resultant  of  fluid  pressure  is  uAz, 
and  is  in  a  Hne  vertically  upward  through  the  centre  of  gravity  of  A. 
The  mean  force  against  which  the  work  is  done  is  therefore  \'J.'Az, 
as  this  is  a  case  in  which  work  is  done  against  a  force  increasing 
from  zero  in  simple  proportion  to  the  space.  Hence  the  work  done 
is  \'Li'Az\  We  see,  therefore,  that  so  far  as  vertical  displacements 
alone  are  ooncerned,  the  equilibrium  is  necessarily  stable,  unless  the 
body  is  wholly  immersed,  when  the  area  of  the  plane  of  flotation 
vanishes,  and  the  equilibrium  is  neutral. 

707.  The  lemma  of  §  704  suggests  that  we  should  take,  as  the 
two  horizontal  axes  of  rotation,  the  principal  axes  of  the  plane  of 
flotation.    Considering  then  rotation  through  an  infinitely  small  angle 


Q  round  one  of  these,  let  G  and  E  be  the  dis;. laced  centres  of  gravity 
of  the  solid,  and  of  the  portion  of  its  \olunie  which  was  immersed 
when  it  was  floating  in  equilibrium,  and  G' ,  E  the  positions  which 
they  then  had;  all  projected  on  the  plane  of  the  diagram  wliich  we 
suppose  to  be  through  /  the  centre  oi  inertia  of  the  plane  of  tlotaticn. 
The  resultant  action  of  gravity  on  the  displaced  l.iody  is  W,  its  weight, 
acting  downwards  through  G;  and  that  uf  the  fluid  pressure  on  it  is 
W^  upwards  through  E  corrected  by  the  amount  (u[)wards)  due  to  tlie 
additional  immersion  of  the  wedge  A/A',  and  the  amount  (down- 
wards) due  to  the  extruded  wedge  BIB.    Hence  the  whole  action  of 


276  ABSTRACT  DYNAMICS. 

gravity  and  fluid  pressure  on  the  displaced  body  is  the  couple  Of 
forces  up  and  down  in  verticals  through  G  and  E,  and  the  correction 
due  to  the  wedges.  This  correction  consists  of  a  force  vertically 
upwards  through  the  centre  of  gravity  of  A'lA,  and  downwards 
through  that  of  BIB\  These  forces  are  equal  [§  704  (i)],  and 
therefore  constitute  a  couple  which  [704  (2)]  has  the  axis  of  the 
displacement  for  its  axis,  and  which  [§  704  (3)]  has  its  rnoment  equal 
to  Owk^A  if  A  be  the  area  of  the  plane  of  flotation,  and  k  its  radhis 
of  gyration  (§  235)  round  the  principal  axis  in  question.  But  since 
GE,  which  was  vertical  (G'E')  in  the  position  of  equilibrium,  .is 
inclined  at  the  infinitely  small  angle  6  to  the  vertical  in  the  displaced 
body,  the  couple  of  forces  TV  in  the  verticals  through  G  and  E  has 
for  moment  IV/id,  if  /i  denote  GE;  and  is  in  a  plane  perpendicular 
to  the  axis,  and  in  the  direction  tending  to  increase  the  displacement, 
when  G  is  above  E.  Hence  the  resultant  action  of  gravity  and  fluid 
pressure  on  the  displaced  body  is  a  couple  whose  moment  is 

(toAk"-  V/h)e,  or  wiAk'-  Vh)9, 

if  F  be  the  volume  immersed.  It  follows  that  when  Ak^>  Vh  the 
equilibrium  is  stable,  so  far  as  this  displacement  alone  is  concerned. 

Also,  since  the  couple  worked  against  in  producing  the  displace- 
ment increases  from  zero  in  simple  proportion  to  the  angle  of  dis- 
placement, its  mean  value  is  half  the  above;  and  therefore  the  whole 
amount  of  work  done  is  equal  to 

\w{Ak^-Vh)Q\ 

708.  If  now  we  consider  a  displacement  compounded  of  a  vertical 
(downwards)  displacement  z,  and  rotations  through  infinitely  small 
angles  Q,  &  round  the  two  horizontal  principal  axes  of  the  plane  of 
flotation,  we  see  (§§  706,  707)  that  the  work  required  to  produce  it  is 
equal  to 


w  [Az'  +  {Ak'  -  Vh)  6'  +  (Ak"  -  V/i)6"], 


and  we  conclude  that,  for  complete  stability  with  reference  to  all  pos- 
sible displacements  of  this  kind,  it  is  necessary  and  sufficient  that 

.      A^'         ,     Ak" 
/i< -y- ,  axia  < —j^  . 

709.  When  the  displacement  is  about  any  axis  through  the  centre 
of  inertia  of  the  plane  of  flotation,  the  resultant  of  fluid  pressures  is 
equal  to  the  weight  of  the  body;  but  it  is  only  when  the  axis  is  a 
principal  axis  of  the  plane  of  flotation  that  this  resultant  is  in  the 
plane  of  displacement.  In  such  a  case  the  point  of  intersection  of 
the  resultant  with  the  line  originally  vertical,  and  through  the  centre 
of  gravity  of  the  body,  is  called  the  Mctaccntre.  And  it  is  obvious, 
from  tlie  above  investigation,  that  for  either  of  these  planes  of  dis- 
placement the  condition  oi"  stable  equilibrium  is  tl;at  the  metacentre 
shall  be  above  the  centre  of  gravity. 

710.  We  shall  conclude  with  the  cons-' deration  of  one  case  of  the 


STATICS  OF  SOLIDS  AND  FLUIDS. 


277 


equilibrium  of  a  revolving  mass  of  fluid  subject  only  to  the  gravitatiorj 
of  its  parts,  which  admits  of  a  very  simple  synthetical  solution,  without 
any  restriction  to  approximate  sphericity;  and  for  which  the  following, 
remarkable  theorem  was  discovered  by  Newton  and  Maclaurin  : — 

711.  An  oblate  ellipsoid  of  revolution,  of  any  given  eccentricity,  is 
a  figure  of  equilibrium  of  a  mass  of  homogeneous  incompressible 
fluid,  rotating  about  an  axis  with  determinate  angular  velocity,  and 
subject  to  no  forces  but  those  of  gravitation  among  its  parts. 

The  angular  velocity  for  a  given  eccentricity  is  independent  of  the 
bulk  of  the  fluid,  and  proportional  to  the  square  root  of  its  density. 

712.  The  proof  of  this  proposition  is  easily  obtained  from  the 
results  already  deduced  with  respect  to  the  attraction  of  an  ellipsoid 
and  the  properties  of  the  free  surface  of  a  fluid. 

We  know,  §538,  that  ii  AFBhe.  a  meridian  section  of  a  homo- 
geneous oblate  spheroid,  AC  the  polar  axis,  CB  an  equatorial  radius, 
and  F  3.x\y  point  on  the  surfoce,  the  attraction  of  the  spheroid  may  be 
resolved  into  two  parts; 
one,  Fp,  perpendicular  to 
the  polar  axis,  and  \ary- 
ing  as  the  ordinate  FM; 
the  other,  Fs,  parallel  to 
the  polar  axis,  and  var}'- 
ing  as  FJV.     These  com- 
ponents   are    not   equal 
when  MF  and  FN  are 
equal,  else  the  resultant 
attraction  at  all  points  in 
the    surface  would   pass 
through   C;   whereas  we 
know  that  it  is  in  some 
such  direction  as  Ff,  cutting  the  radius  FC  b-.ticcen  B  and  (T,  but  a 
a  point  nearer  to  C  than  n  the  foot  of  the  normal  at  F.     Let  tlien 

Fp^a.  FM, 

and     Fs  =  (i.  FN 

uhere  a  and  ft  are  known  constants,  depending  m.erely  on  the  density 
(p),  and  eccentricity  {e),  of  the  spheroid. 

Also,  we  knov.-  by  geometr\-  that  N:  =  {i  -/)  CN. 

Hence;  to  find  the  magnitude  of  a  force  /'/  perpendicular  to  the 
axis  of  the  spheroid,  whicli,  when  compounded  with  the  attraction, 
will  bring  the  resultant  force  into  the  normal  F."  :  m:'kc  />r  =  Fq,  and 
we  must  have 

Fs  ~  FN~  ^  ^     ''  FN "  ^  ^     '"  -*  oj's  ' 


Hence 


Fr  =  {i-r)^Fl, 


2  7  8  ABSTRA  CT  D  YNAMICS. 

or  /V  =  {x-(t-^)f|-P/ 

=  {a-(l-<^)/3}PJ/. 
Now  if  the  spheroid  were  to  rotate  with  angular  velocity  w  about  AC, 
the  centrifugal  force,  §§  39,  42,  225,  would  be  in  the  direction  Pq, 
and  would  amount  to 

ii?PM. 
Hence,  if  we  make 

<o'  =  a-(i-<^)/3, 

the  whole  force-  on  P,  that  is,  the  resultant  of  the  attraction  and 
centrifugal  force,  will  be  in  the  direction  of  the  normal  to  the  sur- 
face, which  is  the  condition  for  the  free  surface  of  a  mass  of  fluid  in 
equilibrium. 

Now,  (§522  of  our  larger  work) 

/n/i"^   •  -1        I- A 
o  =  27r/>(^^^-^3 — sm  '^--p-j, 

y3  =  47rp  0.  -  '^~  sin-'  ^  j  . 

Hence  <o»=  27rp  |C«?Jl^),^ZilZsin-'^- 3  iT  A .  (i) 

This  determines  the  angular  velocity,  and  proves  it  to  be  proportional 
to^p. 

713.     If,  after  Laplace,  we  introduce  instead  of  ^  a  quantity  € 
defined  by  the  equation 


(2) 


(3) 

When  e,  and  therefore  also  c,  is  small,  this  formula  is  most  easily 
calculated  from 

^p  =  T?^"-A«'  +  etc.  (4) 

of  which  the  first  term  is  sufficient  when  we  deal  with  spheroids  so 
little'  oblate  as  the  earth. 

The  following  table  has  been  calculated  by  means  of  these  simpli- 
fied formulae.  The  last  figure  in  each  of  the  four  last  columns  is 
given  to  the  nearest  unit.  The  two  last  columns  will  be  explained  a 
few  sections  later : — 


r     e"        "■ 

e* 

I  + 

or 

e 

-  tan  (sin"' 

')'. 

"'jr-e-- 

the 

expression 

(0 

for 

w^  is  much 
<o^       3  +  e^ 

27rp  ~      c^ 

simplified, 
-tan-'c--^ 

and 

STATICS   OF  SOLIDS  AND   FLUIDS. 


279 


1. 

11. 

111. 

IV. 

V 

if. 

I 

OJ- 

—  when  p— ^•68x10"''. 

1 

6 

ITtp' 

CJ                  '        "^ 

■3  7r/5 

O'l 

9'950 

0-0027 

79,966 

0-C027 

•2 

4-899 

•0107 

39.397 

■Olio 

■3 

3-iSo 

•0243 

26,495 

-0258 

•4 

2-.-9I 

•0436 

19,780 

•0490 

•5 

1732 

•0690 

iS>73a 

•0836 

•6 

io33 

•1007 

13,022 

•1356 

•7 

I  •020 

•I3S7 

11,096 

•2172 

•8 

o"75o 

•I8I6 

9.^97 

•35«8 

•9 

•4S43 

•2203 

8,804 

-6605 

•91 

■455(> 

•2225 

8,759 

•7198      1 

•92 

•4260 

•2241 

8,7-^9 

•7S13 

•93 

■3952 

•2247 

8,718 

•8533 

•94 

•3629 

•2239 

8,732 

•9393 

•95 

•3287 

•2213 

8,783 

I '045 

•96 

•2917 

•2160 

8,891 

I-I79 

•97 

•2506 

•2063 

9,098 

i"359 

•98 

•2030 

•IS90 

9>504 

1-627 

■99 

•1425 

•155 1 

10,490 

2-113 

I'OO 

O'OOOO 

G-QOOO 

CO 

CO 

From  this  we  see  that  the  value  of increases  LTadually  from 

27rp  "^  •' 

zero  to  a  maximum  as  the  eccentricity  c  rises  from  zero  to  about  0-93, 
and  then  (more  quickly)  falls  to  zero  as  the  eccentricity  rises  from 
0-93  to  unity.  The  values  of  the  other  quantities  corresponding  to 
this  ma.ximum  are  given  in  the  table. 

714.  If  the  angular  velocity  exceed  the  value  calculated  from 

r— =-0*2247,  (5) 

27rp 

when  for  p  is  substituted  the  density  of  the  liquid,  equilibrium  is  im- 
possible in  the  form  of  an  ellipsoid  of 'revolution.  If  the  angular 
velocity  fall  short  of  this  limit  there  are  always  two  ellipsoids  of 
revolution  which  satisfy  the  conditions  of  equilibrium.  In  one  of 
these  the  eccentricity  is  greater  than  0-93,  in  the  other  less, 

715.  It  may  be  useful,  for  special  applications,  to  indicate  briefly 
ho-.v  p  is  measured  in  these  formulae.  In  the  detlnitions  of  ^§  476, 
477,  on  which  the  attraction  formulae  are  based,  unit  mass  is  defined 
as  exerting  unit  force  on  unit  mass  at  unit  distance;  and  unit  volume- 
density  is  that  of  a  body  which  has  unit  mass  in  unit  volume.  Hence, 
with  the  foot  as  our  linear  unit,  v/e  have  for  the  earth's  attraction  on 
a  particle  of  unit  mass  at  its  surface 


28o  ABSTRACT  DYNAMICS. 

where  R  is  the  radius  of  the  earth  (supposed  spherical)  in  feet,  and 
o-  its  mean  density,  expressed  in  terms  of  the  unit  just  defined. 
Taking  20,900,000  feet  as  the  value  of  ^,  we  have 

cr  =  0*000000368  =  3'68  X  io~'.  (6) 

As  the  mean  density  of  the  earth  is  somewhere  about  5  '5  times  that 
of  water,  the  density  of  water  in  terms  of  our  present  unit  is 

3'68      _,     ,  _8 
10  '  =  67  X  ic  '. 

716.  The  fourth  column  of  the  table  above  gives  the  time  of  rota- 
tion in  seconds,  corresponding  to  each  value  of  the  eccentricity,  p 
being  assumed  equal  to  the  mean  density  of  the  earth.  For  a  mass 
of  water  these  numbers  must  be  multiplied  by  s/s'S)  a^s  the  time  of 
rotation  to  give  the  same  figure  is  inversely  as  the  square  root  of  the 
density. 

For  a  homogeneous  liquid  mass,  of  the  earth's  mean  density, 
rotating  in  23'^  46™  4*  we  find  ^=0*093,  which  corresponds  to  an 
ellipticity  of  about  ■,^. 

717.  An  interesting  form  of  this  problem,  also  discussed  by  Laplace, 
is  that  in  which  the  moment  of  momentum  and  the  mass  of  the  fluid 
are  given,  not  the  angular  velocity;  and  it  is  required  to  find  what  is 
the  eccentricity  of  the  corresponding  ellipsoid  of  revolution,  the  result 
proving  that  there  can  be  but  one. 

It  is  evident  that  a  mass  of  any  ordinary  liquid  (not  z.  perfect  fluids 
§  684),  if  left  to  itself  in  any  state  of  motion,  must  preserve  unchanged 
its  moment  of  momentum,'  §  202.  But  the  viscosity,  or  iiiternal 
friction,  §  684,  will,  if  the  mass  remain  continuous,  ultimately  destroy 
all  relative  motion  among  its  parts ;  so  that  it  will  ultimately  rotate  as 
a  rigid  solid.  If  the  final  form  be  an  ellipsoid  of  revolution,  we  can 
easily  show  that  there  is  a  single  definite  value  of  its  eccentricity. 
But,  as  it  has  not  yet  been  discovered  whether  there  is  any  other 
form  consistent  with  stable  equilibrium,  we  do  not  know  that  the  mass 
will  necessarily  assume  the  form  of  this  particular  ellipsoid.  '  Nor  in 
fact  do  we  know  whether  even  the  ellipsoid  of  rotation  may  not  beconjie 
an  unstable,  form  if  the  moment  of  momentum  exceed  some  limit  de- 
pending on  the  mass  of  the  fluid.  We  shall  return  to  this  subject  in 
Vol.  II.,  as  it  aff'ords  an  excellent  example  of  that  difficult  and  delicate 
question  Kinetic  Stability,  §  300, 

If  we  call  a  the  equatorial-  semi-axis  of  the  ellipsoid,  e  its  eccen- 
tricity, and  w  its  angular  velocity  of  rotation,  the  given  quantities  are 
the  mass  M-^TrpcfJi-^, 

and  the  moment  of  momentum 

A  =  -j^Trpwo*  Jl-^. 

These  equations,  along  with  (3),  determine  the  three  quantities,  a,  e, 
and  ill. 

Eliminating  a  between  the  two  just  written,  and  expressing  e  as 
before  in  terms  of  t,  we  have 


STATICS   OF  SOLIDS  AXD  FLUIDS.  281 

Tins  frives  —  = -„ 

where  -*  is  a  ^/tr/z  multiple  of  pi.     Substituting  in  771  (2)  we  have 


A=(i^.v{'-^'---'^-:>)' 


Now  the  last  column  of  the  table  in  §  713  shows  that  the  value  of  this 
function  oft  (which  vanishes  with  c)  continually  increases  with  e,  and 
becomes  infinite  when  e  is  infinite.  Hence  there  is  always  one,  and 
only  one,  value  of  f^,  and  therefore  of  <f,  which  satisfies  the  conditions 
of  the  problem. 

718.  All  the  above  results  might  without  much  difficulty  have  been 
obtained  analytically,  by  the  discussion  of  the  equations;  but  we  have 
preferred,  for  once,  to  show  by  an  actual  case  that  numerical  calcula- 
tion may  sometimes  be  of  very  great  use. 

719.  No  one  seems  yet  to  have  attempted  to  solve  the  general 
problem  of  finding  all  the  forms  of  equilibrium  which  a  mass  of 
homogeneous  incompressible  fluid  rotating  with  uniform  angular 
velocity  may  assume.  Unless  the  velocity  be  so  small  that  the  figure 
differs  but  little  from  a  sphere,  the  problem  presents  difficulties  of  an 
exceedingly  formidable  nature.  It  is  therefore  of  some  importance 
to  know  that  we  can  by  a  synthetical  process  show  that  another  form, 
besides  that  of  the  ellipsoid  of  revolution,  may  be  compatible  with 
equilibrium;  viz.  an  ellipsoid  with  three  unequal  axes,  of  which  the 
least  is  the  axis  of  rotation.  This  curious  theorem  was  discovered  by 
Jacobi  in  1834,  and  seems,  simple  as  it  is,  to  have  been  enunciated 
by  him  as  a  challenge  to  the  French  mathematicians'.  For  the  proof 
we  must  refer  to  our  larger  work. 

•  See  a  Paper  by  Liouville,  yournal  de  r£,colt  Polytechnique,  cahier  xxiii.,  foot- 
note to  p.  ■290. 


Vo'i 


APPENDIX. 


KINETICS. 


{a)  In  the  case  of  the  Simple  Pendulum,  a  heavy  particle  is  sus- 
pended from  a  point  by  a  light  inextensible  string.  If  we  suppose  it 
to  be  drawn  aside  from  the  vertical  position  of  equilibrium  and 
allowed  to  fall,  it  will  oscillate  in  one  plane  about  its  lowest  position. 
When  the  string  has  an  inclination  Q  to  the  vertical,  the  weight  mg  of 
the  particle  may  be  resolved  into  mg  cos  0  which  is  balanced  by  the 
tension  of  the  string,  and  mg  sin  B  in  the  direction  of  the  tangent  to 
the  path.  .  If  /  be  the  length  of  the  string,  the  distance  (along  the  arc) 
from  the  position  of  equilibrium  is  IQ. 

Now  if  the  angle  of  oscillation  be  small  (not  above  3°  or  4°  say),  the 
sine  and  the  angle  are  nearly  equal  to  each  other.  Hence  the  acce- 
leration of  the  motion  (which  is  rigorously  ^  sin  &)  may  be  written  ^^. 
Hence  we  have  a  case  of  motion  in  which  the  acceleration  is  ^propor- 
tional to  the  distance  from  a  point  in  the  path,  that  is,  by  §  74,  Simple 
Harmonic  Motion.     The  square  of  the  angular  velocity  in  the  cor- 

responding  circular  motion  is  ,.    , ■  •=  -. ,  and  the  period  of  the 

^  °  displacement     /  ^ 

harmonic  motion  is  therefore  2ir^  /-.     In  the  case  of  the  pendulum, 

\  g 
the  time  of  an  oscillation  from  side  to  side  of  the  vertical  is  usually 

taken — and  is  therefore  tt^  /  -. 
V  g 

(b)  Thus  the  times  of  vibration  of  different  pendulums  are  as  the 
square  roots  of  their  lengths,  for  any  arcs  of  vibration,  provided  only 
these  be  s7nalL 

Also  the  times  of  vibration  of  the  same  pendulum  at  different 
places  are  inversely  as  the  square  roots  of  the  apparent  force  of  gravity 
on  a  unit  mass  at  these  places. 

(c)  It  was  found  experimentally  by  Newton  that  pendulums  of  the 
same  length  vibrate  in  equal  times  at  the  same  place  whatever  be  the 
material  of  which  their  bobs  are  formed.  This  would  evidently  not 
be  the  case  unless  the  weight  were  in  every  case  proportional  to  the 
amount  of  matter  in  the^ob. 


APPENDIX.  283 

{d)  If  the  simple  pendulum  be  slightly  disturbed  in  any  way  from 
its  position  of  equilibrium,  it  will  in  general  describe  very  nearly  an 
ellipse  about  its  lowest  position  as  centre.  This  is  easily  seen  from 
§82. 

{e)  If  the  arc  of  vibration  be  considerable,  the  motion  will  not  be 
simple  harmonic,  and  the  time  of  vibration  will  be  greater  than  that 
above  stated;  since  the  acceleration  being  as  the  sine  of  the  dis- 
placement, is  in  less  and  less  ratio  to  the  displacement  as  the  latter  is 
greater. 

In  this  case,  the  motion  for  any  disturbance  is,  for  pne  revolution, 
approximately  elliptic  as  before;  but  the  ellipse  slowly  turns  round 
the  vertical,  in  the  direction  in  which  the  bob  moves. 

(/)  The  bob  may,  however,  be  so  projected  as  to  revolve  uniformly 
in  a  horizontal  circle,  in  which  case  the  apparatus  is  called  a  Conical 
PenduluDi.  Here  we  have  /sin^  for  the  radius  of  the  circle,  and  the 
force  in  the  direction  of  the  radius  is  T'sin  ^,  where  T'is  the  tension  of 
the  string.  T'cos  B  balances  rng — and  thus  the  force  in  the  radius  of 
the  circle  is  w^tan  Q.     The  square  of  the  angular  velocity  in  the  circle 

is  therefore    ,-?{■,  and   the   time  of  revolution  2it./ ;   or 

/cos  6  V       g 

where  /i  is  the  height  of  the  point  of  suspension  above  the 

plane  of  the  circle.    Thus  all  conical  pendulums  with  the  same  height 
revolve  in  the  same  time. 

(g)  A  rigid  mass  oscillating  about  a  horizontal  axis,  under  the 
action  of  gravity,  constitutes  what  is  called  a  Co>npound  Pendulum. 

When  in  the  course  of  its  motion  the  body  is  inclined  at  any  angle 
B  to  the  position  in  which  it  hangs,  when  in  equilibrium,  it  experiences 
from  gravity,  and  the  resistance  of  the  supports  of  its  axis,  a  couple, 
which  is  easily  seen  to  be  equal  to 

gWh  sin  0, 
where  Wis  the  mass  and  h  the  distance  of  its  centre  of  gravity  from 
the  axis.     This  couple  produces  (§§  232,  235)  acceleration  of  angular 
velocity,  calculated  by  dividing  the  moment  of  the  couple  by  the 
moment  of  inertia  of  the  body.     Hence,  if /denote  the  moment  of 
inertia  about  the  supporting  axis,  the  angular  acceleration  is  equal  to 
gW%\r\Q 
I       ' 
Its  motion  is,  therefore,  identical  (§  (a))  with  that  of  the  simple  pen- 
dulum of  length  equal  to  rrvr. 

If  a  rigid  body  be  supported  about  an  axis,  which  either  passes 
ve'ry  nearly  through  the  centre  of  gravity,  or  is  at  a  very  great  dis- 
tance from  this  point,  the  length  of  the  equivalent  simple  pendulum 
will  be  very  great:  and  it  is  clear  tkit  some  particular  distance  for 
the  point  of  support  from  the  centre  of  gravity  will  render  the  length 


284  APPENDIX. 

of  the  corresponding  simple  pendulum,  and,  therefore,  the  time  of 
vibration,  least  possible. 

To  investigate  these  circumstances  for  all  axes  parallel  to  a  given 
line,  through  the  centre  of  gravity,  let  k  be  the  radius  of  gyration 
round  this  line,  we  have  (§  198), 

and,  therefore,  if  /  be  the  length  of  the  isochronous  simple  pendulum, 
h^  ^  k'  _  {h  -  ky  +  2hk  _  {k  -  ky 

The  second  term  of  the  last  of  these  forms  vanishes  when  /t  =  k,  and 
is  positive  for  all  other  values  of  /i.  The  smallest  value  of  /  is, 
therefore,  2k,  and  this,  the  shortest  length  of  the  isochronous  simple 
pendulum,  is  realized  when  the  axis  of  support  is  at  the  distance  k 
from  the  centre  of  inertia. 

To  find  at  what  distance  //,  from  the  centre  of  inertia  the  axis  must 
be  fixed  to  produce  a  pendulum  isochronous  with  the  simple  pen- 
dulum^  of  given  length  /,  we  have  the  quadratic  equation 

For  the  solution  to  be  possible  we  have  seen  that  /  must  be  greater 
than,  or  at  least  equal  to,  2k.  If  /=  2k,  the  roots  of  this  equation  are 
equal,  ^  being  their  common  value.  For  any  value  of /greater  than 
2k,  the  equation  has  two  real  roots  whose  sum  is  equal  to  /,  and  pro- 
duct equal  to  ^:  hence,  for  any  distance  from  the  centre  of  inertia 
less  than  k,  another  distance  greater  than  k,  which  is  a  third  propor- 
tional to  it  and  k,  gives  the  same  time  of  vibration;  and  the  length  of 
the  simple  pendulum  corresponding  to  either  case,  is  equal  to  the  sum 
of  the  distances  of  the  two  axes  from  the  centre  of  inertia.  This  sum 
is  equal  to  the  distance  between  them  if  the  two  axes  are  in  one 
plane,  through  the  centre  of  inertia,  and  on  opposite  sides  of  this 
point;  and,  therefore,  for  axes  thus  placed,  and  not  equidistant  from 
the  centre  of  inertia,  if  the  times  of  oscillation  of  the  body  when 
successively  supported  upon  them  are  found  to  be  equal,  it  may  be 
inferred  that  the  distance  between  them  is  equal  to  the  length  of 
the  isochronous  simple  pendulum.  As  a  simple  pendulum  exists  only 
in  theor)',  this  proposition  was  taken  advantage  of  by  Kater  for  the 
practical  determination  of  the  force  of  gravity  at  any  station. 

{/:)  A  laiiformly  heavy  and  perfectly  flexible  cord,  placed  in  the  in- 
terior of  a  smooth  tube  in  the  form  of  any  plane  curve,  and  subject  to 
no  external  forces,  will  exert  no  f>ressure  on  ihi  tube  if  it  have  every- 
where the  same  tension,  and  move  with  a  certain  defiiiite  velocity. 

For,  as  in  §  592,  the  statical  pressure  due  to  the  curvature  of  the 

Q 

Tope  per  unit  of  length  is  J"-  (where  o-  is  the  length  of  the  arc  AB 

in  that  figure)  directed  mwards  to  the  centre  of  curvature.    Now,  the 

element  o",  whose  mass  is  ma,  is  moving  in  a  curve  whose  curvature  is 

0  .  mv*a9 

-  with  velocity  v  (suppose).    The  requisite  force  is  — ■ —  =  mv''0; 


APPENDIX.  285 

and  for  unit  of  length  mv''-.     Hence  if  T=mv*  the  theorem  is  true. 

If  we  suppose  a  portion  of  the  tube  to  be  straight,  and  the  whole  ta 
be  moving  with  velocity  v  parallel  to  this  line,  and  against  the  motion 
of  the  cord,  we  shall  have  the  straight  part  of  the  cord  reduced  ta 
rest,  and  an  undulation,  oi  any,  but  unvarying,  form  and  dimensions. 

It 

running  along  it  with  the  linear  velocity  ^  — 

Suppose  the  cord  stretched  by  an  appended  mass  of  ^Fpounds,  and 
suppose  its  length  /  feet  and  its  own  mass  w  pounds.  Then  T=  Wgy 
Im  =  w,  and  the  velocity  of  the  undulation  is 


/ 


— -  feet  per  second. 
w 


iJ)  IV/ien  an  incompressible  liquid  escapes  from  an  orifice,  the  velocity 
is  the  same  as  would  be  acquired  by  falling  from  the  free  surface  to  the 
leiel  of  the  orifice. 

For,  as  we  may  .neglect  (provided  the  vessel  is  large  compared  with 
the  orifice)  the  kinetic  energy  of  the  bulk  of  the  liquid;  the  kinetic 
energy  of  the  escaping  liquid  is  due  to  the  loss  of  potential  energy 
of  the  whole  by  the  depression  of  the  free  surface.  Thus  the  pro- 
position at  once. 

{k)  The  small  oscillations  of  a  liquid  in  a  U  tube  follow  the 
harmonic  law. 

The  tube  being  of  uniform  section  S,  a  depression  of  level,  .x, 
from  the  mean,  on  one  side,  leads  to  a  rise,  x,  on  the  other;  and  if 
the  whole  column  of  fluid  be  of  length  2a,  we  have  the  mass  2aSp 
disturbed  through  a  space  x,  and  acted  on  by  a  force  2Sxgp  tending  to 

bring  it  back.     The  time  of  oscillation  is  therefore  (§  {a}}  2tt  ./- 
ana  is  the  same  for  all  liquids  whatever  be  their  densities. 


INDEX. 


Aberration  gives  hodograph  of  Earth's 

orbit  53 
Abscissae  452 
Absolute  acceleration  64 

—  inotion  63 

• — unit  of  force,  Gauss's,  188;  British  190 
Acceleration,  definition  34;  uniform  32; 

variable  33;  average  33;  angular  57; 

composition  and  resohuion  34,  37 

—  directed  to  a  fixed  centre  45 
, —  in  a  fixed  direction  44 

—  in   logarithmic   spiral  with   uniform 
angular  velocity  about  the  pole  295 

—  in  Simple  Harmonic  Motion  74 

—  in  straight  line,  uniform  43 

—  in  uniform  circular  motion  36,  39,  42 

—  of  momentum  178 

Accurate   measurements,  ne(;essity   for 

A«tion,  Least  279 

—  Maximum  317 
. —  Minimum  31 1 

—  Stationary  281 

—  Varying  282 
Aeolotropic  substance,  an  647 
Alteration  of  latitude  by  hemispherical 

hill,  or  cavity  496;  by  a  crevasse  497 
Ampere's  Theory  of  Electrodynamics 

336 
Amplitude  of  S.  H.  M.  71 
Angle  between  two  lines,  definition  of 

441  Jiote 
Angle  of  repose  473 
Angle,  solid  482 ;    round  a  point  483 ; 

subtended  at  a  point  485 
Angular  acceleration  57 
Angular  measure,  standard  of  357 
Angular  velocity  54;  unit  of  55;  com- 
position of  107,  108 
Anticlastic  surface  120 
Approximate     treatment     of    physical 

questions  391 
Arc,  definition  of  438;    projection  of 

an  439 
Area  of  an  autotomic  plane  circuit  445 
Argument  of  S.  H.  M.  71 
Atmosphere    Homogeneous    695  j    see 

Hontcgeneaui 


Attraction  not  modified  by  interposition 
of  other  matter  474 

—  is  normal  to  equipotential  surfaces 
506 

—  integral  of  normal,  over  a  closed 
surface  510 

—  direct  analytical  calculation  of  494 

—  law  of,  when  a  uniform  spherical 
shell  exerts  no  attraction  on  an  in- 
ternal point  S41 

—  law  of  gravitation  475 

—  of  gravitating,  electric,  or  magnetic 
masses  478 

—  variation  of,  in  crossing  an  attracting 
surface  495 

—  of  a  circular  arc  for  a  particle  at  its 
centre  499 

—  of  a  right  cone  for  a  particle  at  its 
vertex  494  {c) 

—  of  a  cylinder  on  a  particle  in  its  axis 

494  W 
• —  of  a  cylindrical  distribution  of  matter 
508 

—  of  a  uniform  cirailar  disc  on  a  par- 
ticle in  its  axis  494  (a) 

—  of  an  iiifaiite  disc  494 

—  of  two  equal  uniform  discs,  one  posi- 
tive, other  negative  494  (d) 

—  of  an  Ellipsoid  535,  537 ;  of  homo- 
gaieous  ellipsoid  538 ;  Maclaurin's 
Theorem  539 ;  Ivory's  Theorem  540; 
Duhamel's  application  of  Ivory's  The- 
orem 54 1 

—  of  an  ellipsoidal  shell  535 ;  on  an  in- 
ternal particle  536 

—  of  a  -uniform  limited  straight  line  on 
an  external  particle  499  {h) 

—  of  a  nwwitain  on  a  plumb  line 
496  (a) 

■ —  at  the  top  and  the  bottom  of  a  pit 
496  (*) 

—  oi  infinite  parallel  planes  508 

—  of  a  sphere  composed  of  concentric 
shells  of  uniform  density  498 

—  of  a  uniform  sphere  on  an  external 
particle  infinitely  near  its  surface  488 
cor. 

—  of  a  uniform  sphere  534,  54!, 


INDEX, 


287 


Attraction  of  an  uninsulated  sphere 
under  the  influence  of  an  electrified 
particle  493 

— L  of  a  uniform  spherical  shell  on  an 
internal  point  479;  converse  proposi- 
tion 541 

—  of  a  uniform  spherical  shell  on  an 
external  point  48S 

—  of  a  spherical  surface  whose  density 
varies  as  D~^  from  excentric  points 

490  et  seq.;  excentric  point  inside  at- 
tracted point  outside,  and  vice  versa 

491  ;    excentric   and   attracted   point 
both  within  or  both  without  492 

Autotomic  circuit  443 
Average  curvature  14 
Average  stress  645 
Average  velocity  36 
Axiom,    physical   209;    regarding   the 
equilibrium     of    a    non-rigid    body 

584 
Axis  of  a  couple  201 
Axis,  central  579 

Balance,  Coulomb's  Torsion  385 

—  requisites  for  a  good  383 

—  sensibility,  stability  an4  constancy 
of  a  384 

—  statical  principles  of  591 
Balance,  spring  386 
Ballistic  pendulum  263,  272 
Bending  of  a  supported  beam  or  uniform 

bar  015  ;  supported  at  ends  or  middle 

625;  at  ends  and  middle  627 
Bending,  effect  of,  on  cross  section  of 

body  679 
Bifilar  suspension  388 
Body,  motion  of  a  rigid  106 
Body,    a  perfectly  rigid,   defined  393, 

401 
Bramah  Press,  hydrostatic  principle  of 

6S9 
British  system  of  units  of  mass  ^(>i  note 
British  absolute  unit  of  force  190 

Cardioid  105 

Catenary  594;  a  parabola  599;  kinetic 
question  relative  to  59S  ;  inverse  pro- 
blem 600 

Cathetometer  382 

Central  axis  579 

Central  ellipsoid  238 

Centre  of  gravity,  and  centre  of  inertia 
i95>  54^1  5^2;  ccntrobaric  bodies 
542  ;  if  it  exist  is  centre  of  inertia  550 ; 
position  of  in  stable  equilibrium  585, 
in  rocking  stones  586;  of  a  body  in 
equilibrium  about  an  axis  587,  on  a 
fixed  surface  58S;  Pappus'  theoreiTi 
concerning  589 


Centre  of  pressure  688,  703 

Centre  of  mass  or  inertia  195,  582  j 
motion  of  centre  of  inertia  of  a  rigid 
body  232.  5;o;  moments  of  inertia 
of  centrobaric  body  round  axes 
through  centre  of  inertia  551 

Centrobaric  body  542,  proved  possible 
by  Green  543,  properties  of  545; 
centrobaric  shell  547;  centrobaric 
solid  549;  moments  of  inertia  of  a 
centrobaric  body  round  axes  through 
centre  of  inertia  551 

Change  of  velocity  177,  of  momen- 
tum 177 

Characteristic  function,  Hamilton's  283 

Chasles  on  confocal  ellipsoids  537 

Chronometer  367 

Chronoscope  369 

Circuit,  linear  443  ;  autotomic  443 

Circular  measure,  unit  of  357 

Clairault's  formula  for  the  amount  ol 
gravity  at  a  place  187 

Clocks  367 

Closed  curve  443 

—  polygon  443 

Closed  surface,  _//"  yWcr,  over  a  510 

Coarsegrainedness  646 

Coefficient  of  elasticity  265  note,  644 

Coeiticient  of  restitution  265;  of  glass, 
iron,  wool  265 

Comet,  hodograph  of  orbit  of  49 

Component  velocity  29;  acceleration 
37;  of  a  force,  effective  193 

Composition  of  Velocities  31 ;  Accelera- 
tions'>,^;  Simple  Plarmonic  Motions  in 
same  direction  75,  in  different  direc- 
tions 80;  Angular  velocities  107, 
about  axes  meeting  in  a  point  108; 
Rotations  107,  successive  finite  rota- 
tions 109;  Forces  221,  of  two  acting 
on  a  point  419,  422,  special  cases  of 
423  et  seq.;  nearly  conspiring  417, 
nearly  opposed  428,  at  right  angles 
'  429,  of  any  set  of  forces  acting  on  a 
rigid  body  570 ;  Couples  in  same  plane 
or  in  parallel  planes  561,  562,  563,  any 
numl)er,  564  ;  not  in  parallel  planes 
^6^,  any  number  of  506,  and  a  force 
568 

Conipound  pendulum.  Appendix^. 

Compressibility  651 

Conditions  of  equilibrium  of  a  particle 
408;  a  material  point  470;  of  parallel 
forces  558;  of  floating  bodies  702 — 9; 
of  any  number  of  couples  567  et  seq. 

Cone,  orthogonal  and  oblique  section  or 
very  small  486;   solid  angle  of  4S2; 
area   of  segment  cut  from  spherical 
surface  by  a  small  cone  4S7 
Cones  opposite  or  vertical  4S1 


288 


INDEX. 


Confocal  ellipsoids,  corresponding  points 
on  535;  Chasles'  proposition  537 

Conical  pendulum,  Appendix/ 

Conical  surface  480 

Conservation  of  energy  250 

Conservative  system  143 

Constancy  of  a  balance  384 

Constraint  of  a  point  165,  of  a  body 
167  ;  one  degree  of  constraint  of  the 
most  general  character  170 

Contrary  forces  555  note 

Continuity,  equation  of  t6'i 

Conversion  of  units ; — pounds  per  sq. 
inch  to  grammes  per  so.  centimetre 
661 ;  other  units  so-J — 366 

Co-ordinates  452;  propositions  in  co- 
ordinate geometry  459 

Cord  round  cylinder  59J,  603 

Corresponding  points  in  confocal  ellip- 
soids 535 

Cosines,  sum  of  the  squares  of  the  direc- 
tion, of  a  line,  equal  to  unity  460 

Couple  201,  axis  of  loi,  moment  of 
201,  direction  of  560 

-—composition  of  in  same  or  parallel 
planes  561;  any  number  504;  any 
number  not  in  parallel  planes  566 ; 
conditions  of  equilibrium  of  567; 
and  a  force,  composition  of  568  et  seq. 

Curvature  of  a  plane  curve  9 ;  integral 
14;  average  14;  of  a  surface  lao  of 
oblique  sections,  Meunier's  Theorem, 
i^i;  principal, Euler's Theorem  132 

Curvature  of  a  lens,  how  to  measure  381 

Curve,  plane  11;  tortuous  13;  of  double 
curvature  I r;  continuous  35  ;   closed 

443 

Curves  use  of,  in  representing  experi- 
mental results  347 

Cycloid  ()(),  103;  properties  of  104; 
prolate  103;  curtate  103 

D'Alembert's  Principle  230 

Day,  Sidereal  and  Mean  Solar  357 

Degrees  of  freedom  and  constraint  165, 
of  a  point  165,  of  a  body  167;  one 
degree  of  freedom  of  most  general 
character  170 

Density  174  ;  linear,  surface,  volume, 
477  ;  mean  density  of  the  earth  ex- 
pressed in  attraction  units  715 

Developable  surface  125;  practical  con- 
struction of  a,  from  its  edge  133 

Diagonal  scale  372 

Direction  of  motion  8 

Direction  of  rotation,  positive  455 

Direction  cosine  463 ;  sum  of  squares 
of,  equal  to  unity  460 ;  of  the  common 
perpendicular  to  two  lines  464 

DisplacemerU  of  a  plane  figure  in  its 


plane  91,   examples  96;  of  a  rigid 

solid  100 

Dissipation  of  Energy,  instances  147, 
292,  683 

Dissipative  systems  202 

Distortion,  places  of  maximum,  in  a 
cylinder  678 

Distribution  of  the  weight  of  a  solid  on 
points  supporting  it  026 

Double-vpeighing  384 

Duhamel's  application  of  Ivory's  theo- 
rem 541 

Dynamics  t 


Edge  of  regression  13* 

Elastic  body,  a  perfectly  643 

— •  centre  of  a  section  of  a  wire  608 ; 
line  of  elastic  centres,  608;  rotation 
of  a  wire  about  628 

— ■  curve  transmitting  force  and  couple 
619,  properties  of  620;  KirchoflTs  ki- 
netic comparison,  common  pendulum 
and  elastic  asrve  620 

—  solid  equilibrium  of  667 

—  wire  or  fibre  605 

Elasticity,  co-efiicient  of  265  note\  of 
volume  651;  of  figure  651 

Electric  images  528;  definition  530; 
transformation  by  reciprocal  radius 
vectors  531;  electric  image  of  a 
straight  line,  an  angle,  a  circle,  a 
sphere,  a  plane  531 ;  application  to  the 
potential  5325  of  any  distribution  of 
attracting  matter  on  a  spherical  shell 
533;  uniform  shell  eccentrically  re- 
flected 533;  uniform  solid  sphere 
eccentrically  reflected  534 

Elements  of  a  force  184 

Ellipse,  how  to  draw  an  rp 

EUipsoid,  central  237 

Ellipsoid,  attraction  of  a,  53s ;  corre- 
sponding points  on  two  535.;  Ellips- 
oidal shell  defined  535;  attraction  of 
homogeneous  ellipsoidal  shell  on  in- 
ternal point  536;  Potential  constant 
inside  536;  Chasles' Proposition  con* 
ceming  537  j  equijxitential  surfaces  of 
^  537;  Maclaurin's  Theorem  539; 
Ivory's  S40;  comparison  of  the  po- 
tentials of  two  537 

Ellipsoid,  Strain  141;  principal  axes 
of  i4'j 

Empirical  formulae,  use  of  350 

Energy,  kinetic  179;  kinetic  energy  of 
a  system  234;  energy  in  abstract 
dynamics  241,  251;  foundation  of 
the  theory  of  energy  244;  potential 
energy  of  a  conservative  system  245  ; 
conservation  of  E.  250;  inevitable  loss 


INDEX. 


389 


of  energy  of  visible  motion  147 ;  po- 
tential energy  of  a  perfectly  elastic 
body    strainwi    644;    energy    of    a 
strained  isotropic  substance  066 
Epicycloid,    integral   curvature  of  14, 

motion  in  105 
Epoch  in  simple  harmonic  motion  7i 
Equation  of  continuity  i6i ;  integral  and 
differential  163 

Equations  of  mojion  of  any  system  158 

Equilibrium  of  ^partieU,  conditions  of 
408,  470,  on  smooth  and  rough  curves 
and  surfaces  473;  conditions  of  equi- 
librium oi forces  acting  at  a  p>oint  470; 
conditions  of  equilibrium  of  Mr^^y^r^w 
acting  at  a  pomt  584;  graphic  test 
cA  forces  in  equilibrium  4I4;  condi- 
tions for  stable  equilibrium  of  a  body 
585,  rocking  stones  586,  body  move- 
able about  an  axis  587,  bodv  on  a 
fixed  surface  588;  neuiral,  stable,  and 
unstable  equilibrium,  tested  by  the 
principle  of  virtual  velocities  156, 
energy  criterion  of  257  ;  conditions  of 
equilibrium  oi  parallel  forces  558 ;  con- 
ditions of  equilibrium  of  forces  acting 
on  a  rigid  body  576;  equilibrium  of  a 
tton-rtgid  body  not  affected  by  ad- 
ditional fixtures  584,  of  a  flexible  and 
inextensible  cord  i()^ ;  position  of  equi- 
librium o(z  flexible  string  on  a  smooth 
surface  601,  rough  surfaces  6o^  ;  ejui' 
librium  of  elastic  solid  667,  of  incom- 
pressible fluid  completely  filling  rigid 
vessel  696,  under  any  system  of  forces 
697  ;  equilibnum  oi ^floating  body  ia^ 
et  seq. ,  of  a  revolving  mass  of  fluid  710 

Equipotential  surfaces,  examples  of  499, 
505,  526,  of  ellipsoidal  shell  537 

Equivalent  of  pounds  per  square  inch 
in  grammes  per  square  centimetre 
661  ;  other  units  36? — 366. 

Ergometer  389,  Morin's  389 

Experience  3'20 

Experiment  and  observation  374;  n.iles 
for  the  conduct  of  experiment  325 ;  use 
of  empirical  formulae  in  exhibiting 
results  of  experiment  347 

Euler's  theorem  on  curvature  ill,  on 
Impact  •zjG 

Evolute  ^o,  ii 

Flexible  and  inextensible  line,  Kine- 
matics of  a  i6;  flexible  and  inexten- 
sible surface,  flexure  of  125,  general 
property  of  134;  flexible  string  on 
smooth  surface,  position  of  equili- 
brium of  601,  on  rough  surface  603 

Flexure  of  flexible  and  inextensible  sur- 
face 125,  of  a  vv  ire  6oi ;  laws  01  flexuie 


and  torsion  607 ;  axes  of  pure  flextire 
609 ;  case  in  which  the  elastic  central 
line  is  a  normal  axis  of  torsion  609  ; 
where  equal  flexibihty  m  aii  directions 
610;  wire  strained  to  any  given  spiral 
and  twist  6n;  spiral  spring  014; 
principal  axes  of  679  ;  distortion  of  the 
cross  section  of  a  bent  rod  679 

Floating  bodies,  stable  equihbrium  oi, 
lemma  704;  stability  of  705  et  seq.; 
see  Fluid 

Fluid,  properties  of  perfect  401,  684; 
Jluid  pressure  685,  equal  in  all  di- 
rections 686,  proved  by  energy  cri- 
terion 689  ;_/ftttc//lir«jttr;  as  depending 
on  external  forces-  690;  surfcues  of 
equal  pressure  are  perpendicular  to 
Lnes  of  force  69  r,  are  surfaces  of 
equal  density  and  equal  potential  692  ; 
rate  of  increase  of  pressure  694,  in 
a  calm  atmosphere  of  uniform  tem- 
perature 695  (free  surface  in  open 
vessel  is  level  696) ;  resultant  pres- 
sure  on  a  plane  area  702 ;  moment  of 
pressure  702;  loss  of  apparent  weight 
by  immersion  703;  conditioris  of  equi- 
libnum of  a  fluid  completely  filling  a 
closed  vessel  696,  under  non-con- 
servative system  of  forces  697,  im- 
aginary example  699,  actual  case 
701  ;  equilibrium  of  a  floating  body, 
lemma  704,  stability  705,  work  done 
in  a  displacement  705,  metacentn, 
condition  of  its  existence  709;  oblate 
spheroid  is  a  figure  of  equilibrium  of 
a  rotating  incompressible  homoge- 
neous fluid  mass  711;  relation  be- 
tween angular  velocity  of  rotation 
and  density  with  given  ellipticity  712; 
table  of  eccentricities  and  correspond- 
ing angular  velocities  and  moments 
of  momentum  for  a  liquid  of  the 
earth's  mean  density  717;  equilibrium 
of  rotating  ellipsoid  of  three  unequal 
axes  7 1 9 

Fluxion  28 

Forbes's  use  of  Viscous  in  connection 
with  glacier  motion  683 

Force,  moment  of  46,  about  a  point 
199,  source  of  the  idea  of  173,  de- 
fined 183,  specification  of  a  184, 
measure  of  a  185,  meaisurement  of 
224,  by  pendulum  387  ;  force  of  grav- 
ity, Clairault's  formula  for  187,  in 
alDsolute  units  187,  average,  in  Britain 
191;  u'tu  of  force,  gravitation  185, 
absolute  188;  British  absolute  unit 
191;  attraction  unit  of  force  476; 
representation  of  forces  by  lines  192; 
component  of  force  193;  composition 


2go 


inde:\ 


of  forces  ii\,  418,  parallelogram  and 
polygon  of  •272,  true  polygon  of  416, 
triangle  of  410;   forces  through  one 

f)oint,  resultant  of  41a  ;  forces  in  equi- 
ibrium,  graphic  test  of  414;  resultant 
of  three  forces  acting  at  a  point  465  ; 
resolution  of  force  along  three  speci- 
fied lines  468  i  resultant  of  any  nun) her 
of  forces  acting  at  a  point  469 ;  coii' 
dihons  of  equilibrium  of  forces  acting 
at  a  point  470;  resultant  of  forces 
whose  lines  meet  553  ;  two  parallel 
forces  in  a  plane  554,  in  dissimilar 
directions  555,  of  any  number  556, 
not  in  one  plane  557,  conditions  of 
equilibrium  of  538  et  seq. ;  forces  and 
a  couple  568 ;  forces  may  be  reduced 
to  one  force  and  one  couple  570;  re* 
duction  of  forces  to  simplest  system 
571;  parallel  forces  whose  algebraic 
sum  IS  zero  exert  a  directive  action 
only  583 ;  conditions  of  equilibrium 
of  three,  acting  on  a  rigid  body  584; 
conservative  system  of  ■243 

Force  in  terms  of  the  potential  504;  at 
any  point,  due  to  attraction  of  a 
spherical  distribution  of  matter,  a 
cylindrical  distribution,  or  a  disfri- 
bution  in  infinite  planes  508,  where 
It  varies  as  D~'^  509 

Force  of  gravity,  Clairault's  formula  foir 
187,  in  absolute  units  187,  average 
value  in  Britain  191,  in  Edinburgh 
191,  law  of  475 

Force,  line  of,  definition  507,  instance 
of  499  ;  vaiiation  of  intensity  along 
a  508 

Force,  tube  of  508 

Form  of  equilibrium  of  a  rotating  mass 
of  fluid  711 — 719 

Formulae,  use  of  empirical,  in  exhibit- 
ing results  of  experiment  347 

Foucault's  pendulum  87 

Fourier's  theorem  88 

Freedom  of  a  point,,  degrees  of  165;  of 
a  rigid  system  167 

Faction  brake,  White's  300 

Friction,  l^ws  of  statical  403,  kinetic 
404 ;  effect  of  tidal  friction  348  ;  fric- 
cion  of  liquids  varies  as  the  velocity 
292;  friction  of  solids  ^93;  of  a  cord 
round  a  cylinder  591,  603 

Gauss's  absolute  unit    of    force    i88; 

theorem  relating  to  potential  515 
Geodetic  line  i  24,  properties  of  601 
Glacier  Motion,  Forbes's  Viscous  The- 
ory of,  meaning  of  Viscous  in  683 
Gravitation,  law  of  475  -,  potential  503 
Graviiy,   force   of,    Clairault's   formula 


for  187,  at  Edinburgh,  in  Britain 
191,  in  absolute  units  187,  work 
done  against  509 

Gravity,  centre  of,  and  centre  of  inertia 
195,  583;  centre  of  gravity  542  ;  pro- 
perties of  a  body  possessing  a  centre  of 
gravity  544 ;  centre  of  gravity  where 
It  exists  coincides  with  centre  of  inertia 
550;  position  of  centre  of  gravity  in 
a  body  for  stable  equilibnum  585,  m 
rocking  stones  5B6,  in  a  body  with 
erne  point  fixed:  with  two  poimts 
fixed  587,  on  a  surface  588;  Pappus' 
theorem  concerning,  sometimes  called 
Guldinus'  theorem  589 

Green  501;  problem  in  potential  517; 
the  general  problem  of^  electric  in- 
fluence possible  and  determinate  531 

Gyration,  radius  of  235 

Gyroscopes,  motion  of  116 

Hamilton's  Characteristic  Function  183 
Harmonic  motion  69;  simple  harmonic 
motion  70,  amplitude,  argument, 
epoch,  period,  phase,  71,  instances  of 
72;  velocity  in  simple  harmonic  mo- 
tion 73,  cuceleratwn  in  74;  composi- 
tion of  two  simple  harmonic  motions 
in  one  line  75,  examples  77  ;  graphical 
representation  of  simple  harmonic 
motion  in  one  line  79;  composition 
of  simple  harmonic  motion  indifferent 
directions  80;  of  different  kinds  in 
different  directions  84  j  in  two  rect- 
angular directions  85 
Hanon  coal  mine  experiment  498 
Hodograph  49,  of  a  planet  or  comet 
49,  of  a  projectile  50,  of  motion  in 
a  conic  section  51,  of  path  where 
acceleration  is  directed  to  a  fixed 
point  and  varies  as  D~^  61 
Homogeneous  atmosphere  defined  695  : 
height  of  694,  at  Paris,  at  Edinburgh 

695- 
Homogeneous  body  646 
Homogeneous  strain  135;  see  Strtun 
Horsepower  24O 
Hydrodynamics  683  ;  see  Fluid 
Hydrostatics  685;  see  Fluid 
Hyperbola,  how  to  draw  a  (9 
Hypocyoloid  and  hypotrochoid  105 
Hypothesis,  use  of  333 

Image,  electric  538;  see  Electric  im- 
ages 

Impact  359,  duration  of  2l:.();  time  in- 
tegral -263;  ballistic  pendulum  263; 
direct  impact  of  spheres,  Newton's 
experiments  on  265,  loss  of  kinetic 
energy  in  266,  due  to  267,   ca^e  with 


INDEX. 


291 


no  loss  of  kinetic  enerj^  i68;  mo- 
ment of  an  impact  •?;■»;  work  done 
by  J  73;   Euler's  theorem  276 

Impressed  force  183;  see  Force 

Inclination  of  two  given  lines  in  terms 
of  their  direction  cosines  464 

Inerua  i8j 

Inertia,  centre  of  195;  see  Centre  of 
Mass 

Inertia,  moment  of  iqS,  3,^5 — J39,  of 
a  centrobaric  body  551 

Inexterisible  line  16,  surface  ns,  gen- 
eral property  of  134 

Instability  of  motion  300;  instances 
30J,  303,  304 

Interpolation  in  physical  experiments 

350 

Involute  70 

Isotropic  substance  647 

Isotropy,  conditions  fulfilled  in  elastic 
650,  in  one  quality  and  aeolotropy 
in  others  648 

Ivory's  theorem  on  homogeneous  con- 
focal  ellipsoids  540 

Kepler's  first  law  a  consequence  of  ac- 
celeration directed  to  a  fixed  point  45 

Kiloframme  365 

Kinematics  4,  of  a  point  7,  of  an  in- 
extensible  and  flexible  line  16,  of  a 
plane  figfure  9t,  flexible  and  inexten- 
sible  surface  127 

Kinetic  energy  179,  rate  of  change  of 
180;  gain  in  kinetic  energy  equiva- 
lent to  work  done  207 ;  kinetic  energy 
of  a  system  ■234;  loss  of  kinetic  en- 
ergy in  direct  impact  ■266 

Kinetic  foci  310 — 319,  number  of,  in 
any  case  316 

Kinetic  friction  4O4 

Kinetics  1,  3,  4 

Kinetic  stability  300;  kinetic  stability 
or  instability  discriminated  30t ;  cases 
of  kinetic  stability  303,  303,  304; 
kinetic  stability  in  a  circular  orbit 
304;  oscillatory  kinetic  stability  308; 
general  criterion  of  kinetic  stability 
309  ;  motion  on  anticlastic  surfaces  is 
unstable,  synclastic  stable  309 

Kinetic  symmetry  739 

Kirchoffs  kinetic  comparison  between 
twisting  a  wire  and  the  motion  of  a 
pendulum  620 

Latitude    altered    by'  attraction   of   a 
mountain,   or    hcrni>;)hericAl  hill,  or 
cavity  496,  by  a  crevasse  497 
Laws  of  enei^,  dynamical,  •:5  2 
Laws  of  fricLion  403  ;  see  Fiuliort 
Lavrs  of  motion,  history  of  108;    first 


law    iro,    second    217,    third    127, 

Sijholiura  229,  241 
Least  action  279 
Least  squares,  method  of  3^40 
Lemniscate,  integral  curvature  of  r4 
Lengthening  of  a  spiral  spring  due  to 

torsion  OiS 
Level  surface  505 

Limitation  of  dj-namical  problems  391 
Line  density  477 
Line,  expression  for  a    in  co-ordinates 

■.  ""'^ 

Line  of  elastic  centres  remains  un- 
changed in  length  608 ;  see  Elastic 

Line  of  force  def.  507,  instances  of  499, 
variation  of  intensity  along  a  508 

Line,  orthogonal  projection  of  a  442 

Liouid,  effective  moment  of  inertia  of 
075,  no'e 

Locus  of  centre  of  curvature  21 

Longitudinal  vibrations,  velocity  of 
transmission  of  658 

Longitudinal  rigidity  657 

Loss  of  weight  of  body  immersed  in 
fluid  703;  see  Fluid 

Lunar  tides  77 

Machines,  science  of  i 

Maclaurin's  theorem  dn  homogeneous 
confocal  ellipsoids  539 

Mass  174;  measurement  of  175,  214, 
unit  of  190,  365,  476,  715,  I3ritish 
unir  of  190;  mass  v.  weight  I7«, 
186 

—  centre  of  196;  see  Centre  of  Mass 

Matter  173 

Maximum  action  317 

Mean  angular  velocity  58 

Mean  density  174;  of  Earth,  Sche- 
Lallien  experiment  496,  Harton  coal 
mine  experiment  498 

Mean  solar  day  357 

Measure  of  time  358,  371,  of  length 
3^0,  of  surface  363,  of  volume  364, 
of  mass  365,  of  force  366,  of  work 
366,  of  angles  357,  of  pressuic  661 

Measurement  of  force  185,  224,  of 
masses  224 

Mechanical  powers  591 ;  balance  592 

Mechanics  i 

Mechanism  4 

Meiacentre  709;  conditions  for  its  ex- 
istence ;  see  Fluid 

Method  of  least  squares  34O 

Method  of  representing  experimental 
results  347 

Metre  362 

Meunier's  theorem  on  curvatur*  121 

Micrometer  379 

Minimum   action    311;    two   or   more 


r92 


INDEX. 


courses  of  minimum  action  possible- 
3'4 

Modulus  of  elasticity,  Young's  657 ; 
weight  and  length  of  modulus  658; 
specific  modulus  of  isotropic  body 
659  « 

Moment  about  a  point,  of  a  velocity  or 
a  force  46,  representation  of  199,  of 
a  couple  ^oi,  of  an  impact  27^,  of 
pressure  702 

Moment  of  inertia  198,  235,  of  a  cen- 
trobaric  body  551 

Moment  of  momentum  202,  of  a  rigid 
body  232 

Momentum  176,  change  of  177,  accele* 
ration  of  178 

Motion  of  a  material  particle  7;  rela« 
live  motion  63;  simple  harmonic  m.O' 
tion  69;  of  troops  on  suspension 
bridge  78 ;  of  point  of  vibratmg  string 
79 ;  of  a  plane  figure  in  its  own  plane 
19  ;  of  a  rigid  body  about  a  fixed  pomt 
106 ;  general  motion  of  a  rigid  body 
iit;  of  a  screw  in  its  nut  113; 
quantity  of  motion  176;  Newton's 
laws  of  motion  208,  see  Laws;  re- 
sistances to  motion  247;  motion  in 
a  resisting  medium  292,  in  a  logarith- 
mic spiral  295 ;  of  a  system  slightly 
disturbed  from  a  position  of  equilib- 
rium 290 

Neap  tides  77 

Neutral   equilibrium   256;    of  floating 

bodies  705 
Newton's  laws  of  motion  208,  seeZawj; 

experiments  on  impact  265 
Non-conservative  system  298 
Normal  22 
Normal  attraction  over  a  closed  surface* 

integral  of  510 

Oblique  coordinates  453 
Observation  and  experiment  320 
Opposite  cones  481 
Opposite  forces  555,  note 
Ordinates  452 
Orthogonal  projection  442 
Oscillation  in  U  tube,  Appendix  k 

Pcfrallel  forces  in  a  plane,  resultant  of 

two  554,  m  dissimilar  directions  555. 

of  any  number  556,  not  in  one  plane 

,557,   equilibrium  of  55H 
Parallelogram  oi  velocities  31,  offerees 

2ip 
Particle  material  v.  geometrical   point 

7,  181 
Pendulum,  Robins'  ballistic  ^63,   272; 

pendulum  a&^  measurer  of  force  387  ; 


simple  pendulum  Appendix  {a) ;  com- 
pound pendulum  Appendix  (g) 

Perfect  fluid  401,  (^84 

Perfect  solid,  ideal  656 

Perfectly  elastic  body  643 ;  potential 
energy  of  perfectly  elastic  body  held 
strained  644 

Period  of  simple  harmonic  motion  11 

Periodic  disturbance  306 

Periodic  function,  Fourier's  theorem 
regarding  88 

Perpetual  motion  the,  is  impossible  244 

Phase  of  simple  harmonic  motion  71 

Physical  axiom  209 ;  concerning  equi- 
librium 584 

Plane,  osculating  12;  motion  of  plane 
figure  in  its  own  plane  91 

Planet,  path  of  45;  hodograph  of  49,  51, 
6i 

Plasticity  683 

Polar  coordinates  457,  459 

Polygon  plane  11,  gauche  11,  closed 
443,  of  velocities  31,  of  forces  219 

Potential  500;  the  mutual  potential 
energy  of  two  bodies  502,  at  a  point 
503,  force  in  terms  of  potential  504; 
equipotential  surface  505 ;  potential 
due  to  an  attracting  particle  509,  to 
any  mass  509,  potential  cannot  have 
a  maximum  or  minimum  value  at  a 
point  in  free  space  511.  cases  of  this 
515,  5 16;  has  same  value  throughout 
the  interior  as  at  the  surface  of  a  closed 
space  513;  mean  value  of  potential 
throughout  a  sphere  equal  to  the  value 
at  centre  514;  Gauss's  Theorem  515; 
Green's  problem  517;  potential  due 
to  a  uniform  spherical  shell  514,  533  ; 
how  to  distribute  matter  so  as  to  get 
a  given  potential  517 — 521 ,  potential 
due  to  uniform  sphere  534;  due  to 
ellipsoidal  shell  536 

Potential  energy  due  to  work  done  207, 
of  a  conservative  system  245 ;  the 
mutual  potential  energy  of  two  bodies 
502,  of  elastic  solid  held  strained  644 

"Precession  1 17 

Precessional  rotation  116 

Pressure,  centre  of  688.  702  ;  pressure 
at  a  point  in  a  fluid  same  in  every 
direction  6S5,  687.  680;  surfaces  of 
equal  pressure  are  level  surfaces  691 ; 
■whole  pressure  709 

Prmcipal  axes  of  a  strain  144;  ittStram 

I'rincipal  axes  of  mertia  237 

Probable  error  343 

Probable  result  from  a  number  of  obser- 
vations, deduction  of  the  338;  method 
of  least  squares  340;  practical  appli- 
cation 345, 


INDEX. 


293 


Proiect'ile,  path  of  44;  hodograph  50 

Projection  of  areas  200 ;  of  a  point  on  a 
straight  hne  437;  orthogonal  projec- 
tion 447,  of  a  shell  447,  of  any  two 
shells,  of  a  closed  surface  448,  of 
equal  areai  in  iianillel  planes  449,  of 
a  plane  figure  449 

Pulley,  kinematics  of  18 

Pore  strain  159;  see  Strain 

Radius  of  curvature  9 

Radius  of  g}-ration  735 

Regression,  edge  of  132 

Relative  motion  63  et  seq. ;  acceleration 

of  64 

Repo>;e,  angle  of  404,  473 

Residual  phenomena  328 

Resistance  to  motion  247,  250;  varying 
as  the  velocity  in  fluids  293;  to 
change  of  shape,  frictional  683 

Resisting  medium  247 

Resolution  of  velocity  30,  of  forces  43 1, 
geometrical  solution  432,  trigonomet- 
rical solution  433,  in  directions  at 
right  angles  434  ;  application  to  find 
the  resultant  of  a  number  of  forces 
acting  on  a  point  435  ;  resolution  of 
forces  alng  three  specified  liaes  468 

Rest  211 

Restitution,  co-efficient  of  265 

Resultant  velocity  31 ;  resultant  of  forces 
on  a  point  412,  4:9;  three  forces  act- 
ing on  a  point  465,  any  number  470 

Revolving  mass  of  iluid,  equihbrium  of 
7ro;  see  I-luiJ 

Rigid  body,  displacement  of  90,  motion 
of  106,  general  motion  of  112,  rigid 
body  defined  393,  401 

Rigidity  651;  longitudinal  657:  rigidity 
and  resistance  to  compression  655; 
rigidity  as  depending  on  form  677 

Rocking  stones  586 

Rolling  of  bodies  109;  of  curve  upon 
curve  100 

Rolling  motion  118,  119 

Rope  round  cylinder  592,  603 

Rotation,  positive  direction  of  455 

Rotations  about  parallel  axes,  compo» 
sition  of  98;  composition  of  rotation 
and  translation  in  one  plane  99 ;  ro- 
tations of  a  rigid  body,  composition  of 
106;  successive  finite  rotations  109 

Rotation  of  a  wire  round  its  elastic 
central  line  628  ;  sec  Llaitic 

Schehallien  experiment  496 

Scholium  to  law  iii  229,  24! 

ijcrew,  motion  of  a,  in  its  nut  113,  11 4, 

337 
Sea  mile  361 


Section  of  a  small  cone,  oblique  486 

Sensibility  and  stability  of  a  balance  384, 
59i 

Shape,  change  of,  involves  dissipation 
of  energy  683,  247 

Shear,  simple  150,  axes  of  a  152, 
measure  of  a  153,  combined  with  a 
simple  elongation  and  expansion  156 

Shell  def  446 

Siderial  day  358 

Simple  linear  circuit  443 

Simple  harmonic  motion  70,  in  me- 
chanism 72,  composition  of,  in  one 
line  75,  examples  77,  composition  of, 
in  different  directions  80,  of  diflerent 
kinds  in  dilTerent  directions  84,  in  two 
rectangular  directions  85  ;  see  Har- 
vwnic 

Simple  pendulum,  Appendix  (a) 

Simple  shear  150  ;  axes  of  a  shear  152  ; 
ratio  of  a  shear  153,  amount  of  a 
154;  planes  of  no  distortion  in  a  155 

Solar  system,  ultimate  tendency  of  the 
249 

Solar  tides  77 

Solid  angle  4S3;  round  a  point  483; 
subtended  at  a  point  485 

Solid,  elastic  643,  651;  potential  energy 
of  elastic  solid  held  strained  644;  fun- 
damental problems  of  the  mathemati- 
cal theory  of  the  equilibrium  of  an 
elastic  solid  667;  equations  of  internal 
equilibrium  of  668;  St  Venant's  ap- 
plication to  torsion  problems  66g ; 
small  bodies  stronger  than  large  ones 
in  proportion  to  ^heir  weight  683  ; 
imperfectncss  of  elasticity  in  solids 
683 

Space  described  under  uniform  accele- 
ration.in  direction  of  motion  43 

Space,  British  unit  of  190 

Specific  modulus  of  elasticity  689 

Sphere,  attraction  of,  composed  of  con- 
centric shells  of  uniform  density  498  ; 
attraction  of  uniform  sphere  and  po- 
tential due  to  634  ;  see  Attraction  and 
Potaitial 

Spherical  shell,  uniform,  attraction  on 
internal    point    479,    external    point 

488,  on   an   element   of  the   surface 

489,  potential   due  to  533;   see  At- 
traction  »nd  Potential 

Spherometer  },%o 

Spinning  motion  ii3 

Spiral,  motion  in  logarithmic  295,  29^ 

Spiral  springs  3S6,  a>  measurers  of  force 
386,  O14,  kinetic  energyof  O16,  length- 
ening of,  due  to  torsion  618 

Spring  balance;  sec  Spiril  $pni:^i 

Spring  tides  77 


294 


INDEX. 


StaTile  equilibrium  256,  257 ;  sec  Centre 
of  Gravity  and  Floating  Bodies 

Stability  of  motion  300 

Static  friction  404 

Statical  problems,  examples  of  591; 
balance  592;  rod  with  frictionless 
constraint  592 ;  rod  constrained  by 
rough  surfaces  592 ;  block  on  rough 
plane  592  ;  mass  supported  by  rings 
round  rough  post  592;  cord  wound 
round  cylinder  592 

Statics  2,  3,.  of  a  particle  408,  of  a 
rigid  body,  552 

Stationary  action  281 

Straight  beam  infinitely  little  bent 
623 

Strain  135  ;  homogeneous  strain  136  ; 
properties  of  homogeneous  strain  137; 
strain  ellipsoid  141  ;  axes  of  strain 
ellipsoid. 144  ;  elongation  and  change 
of  direction  of  any  line  of  a  body  in 

.  condition  of  strain  145;  distortion 
in  parallel  planes  without  change  of 
volume  148;  simple  shear  150:  axes 
of  a  shear  152,  ratio  of  a  153,  amount 
of  a  154,'  planes  of  no  distortion  m 
a  155,  is  a  simple  elongation  and  ex- 
pansion combined  with  a  shear  156; 
analysis  of  strain  157  ;  pure  strain 
159;  composition  of  pure  160 

Stress  629,  homogeneous  630,  specifi- 
cation of  a  632,  components  of  a 
633,  simple  longitudinal  and  shearing 
stress  633  ;  stress  quadratic  634  ; 
normal  planes  and  axes  of  a  stress 
quadratic  d},^;  varieties  of  stress 
quadratic  636;  laws  of  strain  and 
stress  compared  639 ;  rectangular  ele- 
ments of  strain  and  stress  640  ;  work 
done  by  a  strain  641  ;  a  physical  ap- 
plication 642  ;  stress  produced  by  a 
jingle  longitudinal  stress  653  ;  ratio 
of  lateral  contraction  to  longitudinal 
extension  different  for  different  sub- 
stances 655  ;  stress  required  to  pro- 
duce a  simple  longitudinal  strain  063; 
stress  components  in  terms  of  strain 
for  isotropic  body  664  ;  strain  compo- 
nents in  terms  of  stress  665  ;  funda- 
mental problems  in^  mathematical 
theory  of  equilibrium  of  elastic  solid 
667  ;  equation  of  energy  of  isotropic 
body  666 ;  equations  of  internal 
eauillbrium  668  ;  comparative  strain 
of  similar  bodies  as  depending  oil 
dimensions  682 

St  Vcnant  on  torsion  of  prisms  669 ; 
see  Torsion 

Surface  density  477 

Sudace  of  equilibrium  505  ;  relative  in- 


tensity of  force  at  different  points  of 

a  506 
Surfaces  of  equal  pressure  in  a  fluid  at 

rest  are  also  surfaces  of  equal  potential 

and  equal  density  692 
Symmetry,  kinetic  239 
S)mmetrical  co-ordinates  458,  459 
Synclastic  surface  120 
System,  conservative  2+3 
—  non-conservative  29S 

Tidal  friction,  247,  effect  of  Tides  in 
lengthening  the  period  of  the  Earth's 
rotation  about  iier  axis  248,  249 

Tides  77 

Time,  unit  of  190,  measurement  of  213, 

.358.  371 

Time  integral  262 

Time  of  rotation  of  the  earth  round  its 
axis  increased  by  friction  248 

Time  of  oscillation  of  fluid  in  a  U  tube 
Appendix  ;^;  of  a  simple  pendulum 
Appendix  b,  c,  d,  e,  compound  pen- 
dulum Appendix  g;  wave  running' 
along  a  stretched  cord  Appendix  h 

Tops,  motion  of  spirming  118 

Torsion,  laws  of  007 

Torsion  balance  383 

Torsion  of  a  wire  605  ;  laws  of  607 

Torsion  of-prisms,  St  Venant  on  669, 
lemma  670;  torsion  of  circular  cy- 
linder 671  ;  prism  of  any  shape,  672, 
623  ;  hydrokinetic  analogue  675  ;  con- 
tour lines  for  normal  sections  of 
prisms  &c.  undtr  torsion;'  elliptic 
cylinder,  equilateral  triangular  prism, 
curvilinear  square '  prisms,  square 
prisms :  bars  elliptic,  square,  flat, 
rectangular  676 ;  relation  of  tor- 
sional rigidity  to  flexural  rigidity 
677;  ratio  of  torsional  rigidity  to 
those  of  circular  rods  of  same  mo- 
ment of  inertia,  or  of  same  quantity 
of  material  677;  places  of  greatest 
distortion  in  twiited  prisms  678 

Tortuosity  11 

Tortuous  curve  rr,  13 

Transformation  electrical,  by  reciprocal 
radius  vectors  531 

Transmission  of  force  through  elastic 
solid  629;  transmission  of  homo- 
geneous stress  630;  force  trans- 
mitted across  any  surface  in  elastic 
solid  dii 

Triangle  of  forces  410,  equivalent  to  a 
coujjle  41 1 

Triangle  of  velocities  31 

Trochoid  103 

Tubes  of  force  508 

Turning,  positive  direction  of  455 


hVDEX. 


295 


Unllofm    acceleration    36,    43 1    space 

described  43 
Uniform    circular    motion,   acceleration 

in  .^7;  composition  of  two  86 
Uniform  velocity  23 
Unit  ans^le  55  ;  of  angular  velocity  55  ; 

—  of  angular  measure  357' 

—  of  cubic  measure  364 

—  of  force  1S8,  366,  476 

—  length  3*^10 — 361 

—  mass,  space,  time  190,  365 

—  work  (scienlihc)  204,  gravitation  ■204 

—  surface  363' 

Units,  tables  for  conversion  of  362 — 366, 

661 
Unstable    equilibrium    ■256,    ■257;    see 

Centre  of  Gravity  and  Fluid 

Var)'ing  action  279;  optical  illustration 
286  ;  a  criterion  for  kinetic  stability 

Velocities,  parallelogram,  triangle,  poly- 
gon of  31  ;  examples  of  velocities  41 

Velocity  23.  uniform  23,  variable  26, 
component  29,  resolution  of  29, 
resultant  31,  moment  of  46,  angular 
51,  relative  63,  change  of  177, 
virtual  203,  254 

Velocity  of  a  planet  at  any  point  of  its 
orbit  48;  in  biniple  harmonic  motion 

Velocity  of  escape  of  fluid  from  an  ori- 
fice Appendix  g 

—  of  longitudinal  vibrations  along  a 
rod  658 

—  of  wave  along  stretched  cord  Ap- 
pendix h 


Venant  (St)  on  torsion  669;   see   Tor' 

sion  ' 
Vernier  373 
Vertical  cones  481 
Vibrations    produced    by   impact    220, 

269;    in   a    resisting    medium     293; 

along   stretched    cord    Appendix   /;, 

velocity  of  transmission  of,  through  a 

rod  658 
Virtual   velocity  203,  254,  moment   of 

203 
Viscosity  of  solids  683  ;  of  fluids  683 
Vis  vi\a  179 
Volume,  change  of  involves  dissipation 

of  energy  6^3 
Volume,  density  477,  715 
Volume,  elasticity  of  631 

Weber's  electrical  theory  },^6 

Weight  V.  mass  175  ;  a  measure  of  mass 
,175.,  K^6 

White's  friction  brake  390 

Whole  pressure  on  a  submerged  surface 
702 

Wire,  flexure  of  a  6:2  ;  see  Flexure 

Work  204,  unit  of  204,  against  force 
varying  inversely  as  square  of  distance 
50Q,  independent  of  path  pursued 
under  conservative  system  of  force 
509,  done  in  straining  a  perfectly 
elastic  body  644  ;  transformations  of 
work  207 

\'ard  360 

Young's  modulus  6j7 


K4- 


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